ELFI  Engine for LikelihoodFree InferenceÂ¶
ELFI is a statistical software package for likelihoodfree inference (LFI) such as Approximate Bayesian Computation (ABC). The term LFI refers to a family of inference methods that replace the use of the likelihood function with a data generating simulator function. Other names or related approaches to LFI include simulatorbased inference, approximate Bayesian inference, indirect inference, etc.
ELFI features an easy to use syntax and supports parallelized inference out of the box.
See the quickstart to get started.
ELFI is licensed under BSD3. The source is in GitHub.
Currently implemented LFI methods:Â¶
 ABC rejection sampler
 Sequential Monte Carlo ABC sampler
 Bayesian Optimization for LikelihoodFree Inference (BOLFI) framework
ELFI also has the following non LFI methods:
 Bayesian Optimization
 NoUTurnSampler, a Hamiltonian Monte Carlo MCMC sampler
InstallationÂ¶
ELFI requires Python 3.5 or greater (see below how to install). To install ELFI, simply type in your terminal:
pip install elfi
In some OS you may have to first install numpy
with pip install numpy
. If you don’t
have pip installed, this Python installation guide can guide you through the
process.
Installing Python 3.5Â¶
If you are new to Python, perhaps the simplest way to install it is with Anaconda that manages different Python versions. After installing Anaconda, you can create a Python 3.5. environment with ELFI:
conda create n elfi python=3.5 numpy
source activate elfi
pip install elfi
Optional dependenciesÂ¶
We recommend to install:
graphviz
for drawing graphical models (pip install graphviz
requires Graphviz binaries).
Potential problems with installationÂ¶
ELFI depends on several other Python packages, which have their own dependencies. Resolving these may sometimes go wrong:
 If you receive an error about missing
numpy
, please install it first.  If you receive an error about yaml.load, install
pyyaml
.  On OS X with Anaconda virtual environment say conda install python.app and then use pythonw instead of python.
 Note that ELFI requires Python 3.5 or greater
 In some environments
pip
refers to Python 2.x, and you have to usepip3
to use the Python 3.x version  Make sure your Python installation meets the versions listed in requirements.
Developer installation from sourcesÂ¶
The sources for ELFI can be downloaded from the Github repo.
You can either clone the public repository:
git clone https://github.com/elfidev/elfi.git
Or download the development tarball:
curl OL https://github.com/elfidev/elfi/tarball/dev
Note that for development it is recommended to base your work on the dev branch instead of master.
Once you have a copy of the source, go to the folder and type:
pip install e .
This will install ELFI along with its default requirements. Note that the dot in the end means the current folder.
Good to knowÂ¶
Here we describe some important concepts related to ELFI. These will help in understanding how to implement custom operations (such as simulators or summaries) and can potentially save the user from some pitfalls.
ELFI modelÂ¶
In ELFI, the priors, simulators, summaries, distances, etc. are called operations. ELFI provides a convenient syntax of combining these operations into a network that is called an ElfiModel, where each node represents an operation. Basically, the ElfiModel is a description of how different quantities needed in the inference are to be generated. The structure of the network is a directed acyclic graph (DAG).
OperationsÂ¶
Operations are functions (or more generally Python callables) in the nodes of the
ELFI model. Those nodes that deal directly with data, e.g. priors, simulators,
summaries and distances should return a numpy array of length batch_size
that contains
their output.
If your operation does not produce data wrapped to numpy arrays, you can use the
elfi.tools.vectorize tool to achieve that. Note that sometimes it is required to specify
which arguments to the vectorized function will be constants and at other times also
specify the datatype (when automatic numpy array conversion does not produce desired
result). It is always good to check that the output is sane using the node.generate
method.
Reusing dataÂ¶
The OutputPool object can be used to store the outputs of any node in the graph. Note however that changing a node in the model will change the outputs of it’s child nodes. In Rejection sampling you can alter the child nodes of the nodes in the OutputPool and safely reuse the OutputPool with the modified model. This is especially handy when saving the simulations and trying out different summaries. BOLFI allows you to use the stored data as initialization data.
However passing a modified model with the OutputPool of the original model will produce biased results in other algorithms besides Rejection sampling. This is because more advanced algorithms learn from previous results. If the results change in some way, so will also the following parameter values and thus also their simulations and other nodes that depend on them. The Rejection sampling does not suffer from this because it always samples new parameter values directly from the priors, and therefore modified distance outputs have no effect to the parameter values of any later simulations.
QuickstartÂ¶
First ensure you have installed Python 3.5 (or greater) and ELFI. After installation you can start using ELFI:
ELFI includes an easy to use generative modeling syntax, where the generative model is specified as a directed acyclic graph (DAG). Letâ€™s create two prior nodes:
mu = elfi.Prior('uniform', 2, 4)
sigma = elfi.Prior('uniform', 1, 4)
The above would create two prior nodes, a uniform distribution from 2
to 2 for the mean mu
and another uniform distribution from 1 to 5
for the standard deviation sigma
. All distributions from
scipy.stats
are available.
For likelihoodfree models we typically need to define a simulator and summary statistics for the data. As an example, lets define the simulator as 30 draws from a Gaussian distribution with a given mean and standard deviation. Let’s use mean and variance as our summaries:
import scipy.stats as ss
import numpy as np
def simulator(mu, sigma, batch_size=1, random_state=None):
mu, sigma = np.atleast_1d(mu, sigma)
return ss.norm.rvs(mu[:, None], sigma[:, None], size=(batch_size, 30), random_state=random_state)
def mean(y):
return np.mean(y, axis=1)
def var(y):
return np.var(y, axis=1)
Letâ€™s now assume we have some observed data y0
(here we just create
some with the simulator):
# Set the generating parameters that we will try to infer
mean0 = 1
std0 = 3
# Generate some data (using a fixed seed here)
np.random.seed(20170525)
y0 = simulator(mean0, std0)
print(y0)
[[ 3.7990926 1.49411834 0.90999905 2.46088006 0.10696721 0.80490023
0.7413415 5.07258261 0.89397268 3.55462229 0.45888389 3.31930036
0.55378741 3.00865492 1.59394854 3.37065996 5.03883749 2.73279084
6.10128027 5.09388631 1.90079255 1.7161259 3.86821266 0.4963219
1.64594033 2.51620566 0.83601666 2.68225112 2.75598375 6.02538356]]
Now we have all the components needed. Letâ€™s complete our model by adding the simulator, the observed data, summaries and a distance to our model:
# Add the simulator node and observed data to the model
sim = elfi.Simulator(simulator, mu, sigma, observed=y0)
# Add summary statistics to the model
S1 = elfi.Summary(mean, sim)
S2 = elfi.Summary(var, sim)
# Specify distance as euclidean between summary vectors (S1, S2) from simulated and
# observed data
d = elfi.Distance('euclidean', S1, S2)
If you have graphviz
installed to your system, you can also
visualize the model:
# Plot the complete model (requires graphviz)
elfi.draw(d)
Note
The automatic naming of nodes may not work in all environments e.g. in interactive Python shells. You can alternatively provide a name argument for the nodes, e.g. S1 = elfi.Summary(mean, sim, name='S1')
.
We can try to infer the true generating parameters mean0
and
std0
above with any of ELFIâ€™s inference methods. Letâ€™s use ABC
Rejection sampling and sample 1000 samples from the approximate
posterior using threshold value 0.5:
rej = elfi.Rejection(d, batch_size=10000, seed=30052017)
res = rej.sample(1000, threshold=.5)
print(res)
Method: Rejection
Number of samples: 1000
Number of simulations: 120000
Threshold: 0.492
Sample means: mu: 0.748, sigma: 3.1
Let’s plot also the marginal distributions for the parameters:
import matplotlib.pyplot as plt
res.plot_marginals()
plt.show()
APIÂ¶
This file describes the classes and methods available in ELFI.
Modelling APIÂ¶
Below is the API for creating generative models.
elfi.ElfiModel ([name, observed, source_net]) 
A container for the inference model. 
General model nodes
elfi.Constant (value, **kwargs) 
A node holding a constant value. 
elfi.Operation (fn, *parents, **kwargs) 
A generic deterministic operation node. 
elfi.RandomVariable (distribution, *params[, ...]) 
A node that draws values from a random distribution. 
LFI nodes
elfi.Prior (distribution, *params[, size]) 
A parameter node of an ELFI graph. 
elfi.Simulator (fn, *params, **kwargs) 
A simulator node of an ELFI graph. 
elfi.Summary (fn, *parents, **kwargs) 
A summary node of an ELFI graph. 
elfi.Discrepancy (discrepancy, *parents, **kwargs) 
A discrepancy node of an ELFI graph. 
elfi.Distance (distance, *summaries, **kwargs) 
A convenience class for the discrepancy node. 
Other
elfi.new_model ([name, set_default]) 
Create a new ElfiModel instance. 
elfi.load_model (name[, prefix, set_default]) 
Load the pickled ElfiModel. 
elfi.get_default_model () 
Return the current default ElfiModel instance. 
elfi.set_default_model ([model]) 
Set the current default ElfiModel instance. 
elfi.draw (G[, internal, param_names, ...]) 
Draw the ElfiModel. 
Inference APIÂ¶
Below is a list of inference methods included in ELFI.
elfi.Rejection (model[, discrepancy_name, ...]) 
Parallel ABC rejection sampler. 
elfi.SMC (model[, discrepancy_name, output_names]) 
Sequential Monte Carlo ABC sampler. 
elfi.BayesianOptimization (model[, ...]) 
Bayesian Optimization of an unknown target function. 
elfi.BOLFI (model[, target_name, bounds, ...]) 
Bayesian Optimization for LikelihoodFree Inference (BOLFI). 
Result objects
OptimizationResult (x_min, **kwargs) 
Base class for results from optimization. 
Sample (method_name, outputs, parameter_names) 
Sampling results from inference methods. 
SmcSample (method_name, outputs, ...) 
Container for results from SMCABC. 
BolfiSample (method_name, chains, ...) 
Container for results from BOLFI. 
Postprocessing
elfi.adjust_posterior (sample, model, ...[, ...]) 
Adjust the posterior using local regression. 
LinearAdjustment (**kwargs) 
Regression adjustment using a local linear model. 
Diagnostics
elfi.TwoStageSelection (simulator, fn_distance) 
Perform the summarystatistics selection proposed by Nunes and Balding (2010). 
Acquisition methods
LCBSC (*args[, delta]) 
Lower Confidence Bound Selection Criterion. 
MaxVar ([quantile_eps]) 
The maximum variance acquisition method. 
RandMaxVar ([quantile_eps, sampler, ...]) 
The randomised maximum variance acquisition method. 
ExpIntVar ([quantile_eps, integration, ...]) 
The Expected Integrated Variance (ExpIntVar) acquisition method. 
UniformAcquisition (model[, prior, n_inits, ...]) 
Acquisition from uniform distribution. 
OtherÂ¶
Data pools
elfi.OutputPool ([outputs, name, prefix]) 
Store node outputs to dictionarylike stores. 
elfi.ArrayPool ([outputs, name, prefix]) 
OutputPool that uses binary .npy files as default stores. 
Module functions
elfi.get_client () 
Get the current ELFI client instance. 
elfi.set_client ([client]) 
Set the current ELFI client instance. 
Tools
elfi.tools.vectorize (operation[, constants, ...]) 
Vectorize an operation. 
elfi.tools.external_operation (command[, ...]) 
Wrap an external command as a Python callable (function). 
Class documentationsÂ¶
Modelling API classesÂ¶

class
elfi.
ElfiModel
(name=None, observed=None, source_net=None)[source]Â¶ A container for the inference model.
The ElfiModel is a directed acyclic graph (DAG), whose nodes represent parts of the inference task, for example the parameters to be inferred, the simulator or a summary statistic.
Initialize the inference model.
Parameters:  name (str, optional) –
 observed (dict, optional) – Observed data with node names as keys.
 source_net (nx.DiGraph, optional) –
 set_current (bool, optional) – Sets this model as the current (default) ELFI model

generate
(batch_size=1, outputs=None, with_values=None)[source]Â¶ Generate a batch of outputs using the global numpy seed.
This method is useful for testing that the ELFI graph works.
Parameters:  batch_size (int, optional) –
 outputs (list, optional) –
 with_values (dict, optional) – You can specify values for nodes to use when generating data

get_reference
(name)[source]Â¶ Return a new reference object for a node in the model.
Parameters: name (str) –

classmethod
load
(name, prefix)[source]Â¶ Load the pickled ElfiModel.
Assumes there exists a file “name.pkl” in the current directory.
Parameters:  name (str) – Name of the model file to load (without the .pkl extension).
 prefix (str) – Path to directory where the model file is located, optional.
Returns: Return type:

name
Â¶ Return name of the model.

observed
Â¶ Return the observed data for the nodes in a dictionary.

parameter_names
Â¶ Return a list of model parameter names in an alphabetical order.

save
(prefix=None)[source]Â¶ Save the current model to pickled file.
Parameters: prefix (str, optional) – Path to the directory under which to save the model. Default is the current working directory.

update_node
(name, updating_name)[source]Â¶ Update node with updating_node in the model.
The node with name name gets the state (operation), parents and observed data (if applicable) of the updating_node. The updating node is then removed from the graph.
Parameters:  name (str) –
 updating_name (str) –

class
elfi.
Constant
(value, **kwargs)[source]Â¶ A node holding a constant value.
Initialize a node holding a constant value.
Parameters: value – The constant value of the node. 
become
(other_node)Â¶ Make this node become the other_node.
The children of this node will be preserved.
Parameters: other_node (NodeReference) –

generate
(batch_size=1, with_values=None)Â¶ Generate output from this node.
Useful for testing.
Parameters:  batch_size (int, optional) –
 with_values (dict, optional) –

parents
Â¶ Get all positional parent nodes (inputs) of this node.
Returns: parents – List of positional parents Return type: list

reference
(name, model)Â¶ Construct a reference for an existing node in the model.
Parameters:  name (string) – name of the node
 model (ElfiModel) –
Returns: Return type: NodePointer instance

state
Â¶ Return the state dictionary of the node.


class
elfi.
Operation
(fn, *parents, **kwargs)[source]Â¶ A generic deterministic operation node.
Initialize a node that performs an operation.
Parameters: fn (callable) – The operation of the node. 
become
(other_node)Â¶ Make this node become the other_node.
The children of this node will be preserved.
Parameters: other_node (NodeReference) –

generate
(batch_size=1, with_values=None)Â¶ Generate output from this node.
Useful for testing.
Parameters:  batch_size (int, optional) –
 with_values (dict, optional) –

parents
Â¶ Get all positional parent nodes (inputs) of this node.
Returns: parents – List of positional parents Return type: list

reference
(name, model)Â¶ Construct a reference for an existing node in the model.
Parameters:  name (string) – name of the node
 model (ElfiModel) –
Returns: Return type: NodePointer instance

state
Â¶ Return the state dictionary of the node.


class
elfi.
RandomVariable
(distribution, *params, size=None, **kwargs)[source]Â¶ A node that draws values from a random distribution.
Initialize a node that represents a random variable.
Parameters:  distribution (str or scipylike distribution object) –
 params (params of the distribution) –
 size (int, tuple or None, optional) – Output size of a single random draw.

become
(other_node)Â¶ Make this node become the other_node.
The children of this node will be preserved.
Parameters: other_node (NodeReference) –

static
compile_operation
(state)[source]Â¶ Compile a callable operation that samples the associated distribution.
Parameters: state (dict) –

distribution
Â¶ Return the distribution object.

generate
(batch_size=1, with_values=None)Â¶ Generate output from this node.
Useful for testing.
Parameters:  batch_size (int, optional) –
 with_values (dict, optional) –

parents
Â¶ Get all positional parent nodes (inputs) of this node.
Returns: parents – List of positional parents Return type: list

reference
(name, model)Â¶ Construct a reference for an existing node in the model.
Parameters:  name (string) – name of the node
 model (ElfiModel) –
Returns: Return type: NodePointer instance

size
Â¶ Return the size of the output from the distribution.

state
Â¶ Return the state dictionary of the node.

class
elfi.
Prior
(distribution, *params, size=None, **kwargs)[source]Â¶ A parameter node of an ELFI graph.
Initialize a Prior.
Parameters:  distribution (str, object) – Any distribution from scipy.stats, either as a string or an object. Objects must implement at least an rvs method with signature rvs(*parameters, size, random_state). Can also be a custom distribution object that implements at least an rvs method. Many of the algorithms also require the pdf and logpdf methods to be available.
 size (int, tuple or None, optional) – Output size of a single random draw.
 params – Parameters of the prior distribution
 kwargs –
Notes
The parameters of the scipy distributions (typically loc and scale) must be given as positional arguments.
Many algorithms (e.g. SMC) also require a pdf method for the distribution. In general the definition of the distribution is a subset of scipy.stats.rv_continuous.
Scipy distributions: https://docs.scipy.org/doc/scipy0.19.0/reference/stats.html

become
(other_node)Â¶ Make this node become the other_node.
The children of this node will be preserved.
Parameters: other_node (NodeReference) –

compile_operation
(state)Â¶ Compile a callable operation that samples the associated distribution.
Parameters: state (dict) –

distribution
Â¶ Return the distribution object.

generate
(batch_size=1, with_values=None)Â¶ Generate output from this node.
Useful for testing.
Parameters:  batch_size (int, optional) –
 with_values (dict, optional) –

parents
Â¶ Get all positional parent nodes (inputs) of this node.
Returns: parents – List of positional parents Return type: list

reference
(name, model)Â¶ Construct a reference for an existing node in the model.
Parameters:  name (string) – name of the node
 model (ElfiModel) –
Returns: Return type: NodePointer instance

size
Â¶ Return the size of the output from the distribution.

state
Â¶ Return the state dictionary of the node.

class
elfi.
Simulator
(fn, *params, **kwargs)[source]Â¶ A simulator node of an ELFI graph.
Simulator nodes are stochastic and may have observed data in the model.
Initialize a Simulator.
Parameters:  fn (callable) – Simulator function with a signature sim(*params, batch_size, random_state)
 params – Input parameters for the simulator.
 kwargs –

become
(other_node)Â¶ Make this node become the other_node.
The children of this node will be preserved.
Parameters: other_node (NodeReference) –

generate
(batch_size=1, with_values=None)Â¶ Generate output from this node.
Useful for testing.
Parameters:  batch_size (int, optional) –
 with_values (dict, optional) –

parents
Â¶ Get all positional parent nodes (inputs) of this node.
Returns: parents – List of positional parents Return type: list

reference
(name, model)Â¶ Construct a reference for an existing node in the model.
Parameters:  name (string) – name of the node
 model (ElfiModel) –
Returns: Return type: NodePointer instance

state
Â¶ Return the state dictionary of the node.

class
elfi.
Summary
(fn, *parents, **kwargs)[source]Â¶ A summary node of an ELFI graph.
Summary nodes are deterministic operations associated with the observed data. if their parents hold observed data it will be automatically transformed.
Initialize a Summary.
Parameters:  fn (callable) – Summary function with a signature summary(*parents)
 parents – Input data for the summary function.
 kwargs –

become
(other_node)Â¶ Make this node become the other_node.
The children of this node will be preserved.
Parameters: other_node (NodeReference) –

generate
(batch_size=1, with_values=None)Â¶ Generate output from this node.
Useful for testing.
Parameters:  batch_size (int, optional) –
 with_values (dict, optional) –

parents
Â¶ Get all positional parent nodes (inputs) of this node.
Returns: parents – List of positional parents Return type: list

reference
(name, model)Â¶ Construct a reference for an existing node in the model.
Parameters:  name (string) – name of the node
 model (ElfiModel) –
Returns: Return type: NodePointer instance

state
Â¶ Return the state dictionary of the node.

class
elfi.
Discrepancy
(discrepancy, *parents, **kwargs)[source]Â¶ A discrepancy node of an ELFI graph.
This class provides a convenience node for custom distance operations.
Initialize a Discrepancy.
Parameters:  discrepancy (callable) – Signature of the discrepancy function is of the form: discrepancy(summary_1, summary_2, ..., observed), where summaries are arrays containing batch_size simulated values and observed is a tuple (observed_summary_1, observed_summary_2, ...). The callable object should return a vector of discrepancies between the simulated summaries and the observed summaries.
 *parents – Typically the summaries for the discrepancy function.
 **kwargs –
See also
elfi.Distance
 creating common distance discrepancies.

become
(other_node)Â¶ Make this node become the other_node.
The children of this node will be preserved.
Parameters: other_node (NodeReference) –

generate
(batch_size=1, with_values=None)Â¶ Generate output from this node.
Useful for testing.
Parameters:  batch_size (int, optional) –
 with_values (dict, optional) –

parents
Â¶ Get all positional parent nodes (inputs) of this node.
Returns: parents – List of positional parents Return type: list

reference
(name, model)Â¶ Construct a reference for an existing node in the model.
Parameters:  name (string) – name of the node
 model (ElfiModel) –
Returns: Return type: NodePointer instance

state
Â¶ Return the state dictionary of the node.

class
elfi.
Distance
(distance, *summaries, **kwargs)[source]Â¶ A convenience class for the discrepancy node.
Initialize a distance node of an ELFI graph.
This class contains many common distance implementations through scipy.
Parameters:  distance (str, callable) –
If string it must be a valid metric from scipy.spatial.distance.cdist.
Is a callable, the signature must be distance(X, Y), where X is a n x m array containing n simulated values (summaries) in rows and Y is a 1 x m array that contains the observed values (summaries). The callable should return a vector of distances between the simulated summaries and the observed summaries.
 *summaries – Summary nodes of the model.
 **kwargs – Additional parameters may be required depending on the chosen distance. See the scipy documentation. (The support is not exhaustive.) ELFIrelated kwargs are passed on to elfi.Discrepancy.
Examples
>>> d = elfi.Distance('euclidean', summary1, summary2...) # doctest: +SKIP
>>> d = elfi.Distance('minkowski', summary, p=1) # doctest: +SKIP
Notes
Your summaries need to be scalars or vectors for this method to work. The summaries will be first stacked to a single 2D array with the simulated summaries in the rows for every simulation and the distance is taken row wise against the corresponding observed summary vector.
Scipy distances: https://docs.scipy.org/doc/scipy/reference/generated/generated/scipy.spatial.distance.cdist.html # noqa
See also
elfi.Discrepancy
 A general discrepancy node

become
(other_node)Â¶ Make this node become the other_node.
The children of this node will be preserved.
Parameters: other_node (NodeReference) –

generate
(batch_size=1, with_values=None)Â¶ Generate output from this node.
Useful for testing.
Parameters:  batch_size (int, optional) –
 with_values (dict, optional) –

parents
Â¶ Get all positional parent nodes (inputs) of this node.
Returns: parents – List of positional parents Return type: list

reference
(name, model)Â¶ Construct a reference for an existing node in the model.
Parameters:  name (string) – name of the node
 model (ElfiModel) –
Returns: Return type: NodePointer instance

state
Â¶ Return the state dictionary of the node.
 distance (str, callable) –
Other

elfi.
new_model
(name=None, set_default=True)[source]Â¶ Create a new
ElfiModel
instance.In addition to making a new ElfiModel instance, this method sets the new instance as the default for new nodes.
Parameters:  name (str, optional) –
 set_default (bool, optional) – Whether to set the newly created model as the current model.

elfi.
load_model
(name, prefix=None, set_default=True)[source]Â¶ Load the pickled ElfiModel.
Assumes there exists a file “name.pkl” in the current directory. Also sets the loaded model as the default model for new nodes.
Parameters:  name (str) – Name of the model file to load (without the .pkl extension).
 prefix (str) – Path to directory where the model file is located, optional.
 set_default (bool, optional) – Set the loaded model as the default model. Default is True.
Returns: Return type:

elfi.
get_default_model
()[source]Â¶ Return the current default
ElfiModel
instance.New nodes will be added to this model by default.

elfi.
set_default_model
(model=None)[source]Â¶ Set the current default
ElfiModel
instance.New nodes will be placed the given model by default.
Parameters: model (ElfiModel, optional) – If None, creates a new ElfiModel
.

elfi.
draw
(G, internal=False, param_names=False, filename=None, format=None)Â¶ Draw the ElfiModel.
Parameters:  G (nx.DiGraph or ElfiModel) – Graph or model to draw
 internal (boolean, optional) – Whether to draw internal nodes (starting with an underscore)
 param_names (bool, optional) – Show param names on edges
 filename (str, optional) – If given, save the dot file into the given filename.
 format (str, optional) – format of the file
Notes
Requires the optional ‘graphviz’ library.
Returns: A GraphViz dot representation of the model. Return type: dot
Inference API classesÂ¶

class
elfi.
Rejection
(model, discrepancy_name=None, output_names=None, **kwargs)[source]Â¶ Parallel ABC rejection sampler.
For a description of the rejection sampler and a general introduction to ABC, see e.g. Lintusaari et al. 2016.
References
Lintusaari J, Gutmann M U, Dutta R, Kaski S, Corander J (2016). Fundamentals and Recent Developments in Approximate Bayesian Computation. Systematic Biology. http://dx.doi.org/10.1093/sysbio/syw077.
Initialize the Rejection sampler.
Parameters:  model (ElfiModel or NodeReference) –
 discrepancy_name (str, NodeReference, optional) – Only needed if model is an ElfiModel
 output_names (list, optional) – Additional outputs from the model to be included in the inference result, e.g. corresponding summaries to the acquired samples
 kwargs – See InferenceMethod

batch_size
Â¶ Return the current batch_size.

extract_result
()[source]Â¶ Extract the result from the current state.
Returns: result Return type: Sample

infer
(*args, vis=None, **kwargs)Â¶ Set the objective and start the iterate loop until the inference is finished.
See the other arguments from the set_objective method.
Returns: result Return type: Sample

iterate
()Â¶ Advance the inference by one iteration.
This is a way to manually progress the inference. One iteration consists of waiting and processing the result of the next batch in succession and possibly submitting new batches.
Notes
If the next batch is ready, it will be processed immediately and no new batches are submitted.
New batches are submitted only while waiting for the next one to complete. There will never be more batches submitted in parallel than the max_parallel_batches setting allows.
Returns: Return type: None

parameter_names
Â¶ Return the parameters to be inferred.

plot_state
(**options)[source]Â¶ Plot the current state of the inference algorithm.
This feature is still experimental and only supports 1d or 2d cases.

pool
Â¶ Return the output pool of the inference.

prepare_new_batch
(batch_index)Â¶ Prepare values for a new batch.
ELFI calls this method before submitting a new batch with an increasing index batch_index. This is an optional method to override. Use this if you have a need do do preparations, e.g. in Bayesian optimization algorithm, the next acquisition points would be acquired here.
If you need provide values for certain nodes, you can do so by constructing a batch dictionary and returning it. See e.g. BayesianOptimization for an example.
Parameters: batch_index (int) – next batch_index to be submitted Returns: batch – Keys should match to node names in the model. These values will override any default values or operations in those nodes. Return type: dict or None

sample
(n_samples, *args, **kwargs)Â¶ Sample from the approximate posterior.
See the other arguments from the set_objective method.
Parameters:  n_samples (int) – Number of samples to generate from the (approximate) posterior
 *args –
 **kwargs –
Returns: result
Return type:

seed
Â¶ Return the seed of the inference.

set_objective
(n_samples, threshold=None, quantile=None, n_sim=None)[source]Â¶ Set objective for inference.
Parameters:  n_samples (int) – number of samples to generate
 threshold (float) – Acceptance threshold
 quantile (float) – In between (0,1). Define the threshold as the pquantile of all the simulations. n_sim = n_samples/quantile.
 n_sim (int) – Total number of simulations. The threshold will be the n_samples smallest discrepancy among n_sim simulations.

class
elfi.
SMC
(model, discrepancy_name=None, output_names=None, **kwargs)[source]Â¶ Sequential Monte Carlo ABC sampler.
Initialize the SMCABC sampler.
Parameters:  model (ElfiModel or NodeReference) –
 discrepancy_name (str, NodeReference, optional) – Only needed if model is an ElfiModel
 output_names (list, optional) – Additional outputs from the model to be included in the inference result, e.g. corresponding summaries to the acquired samples
 kwargs – See InferenceMethod

batch_size
Â¶ Return the current batch_size.

current_population_threshold
Â¶ Return the threshold for current population.

extract_result
()[source]Â¶ Extract the result from the current state.
Returns: Return type: SmcSample

infer
(*args, vis=None, **kwargs)Â¶ Set the objective and start the iterate loop until the inference is finished.
See the other arguments from the set_objective method.
Returns: result Return type: Sample

iterate
()Â¶ Advance the inference by one iteration.
This is a way to manually progress the inference. One iteration consists of waiting and processing the result of the next batch in succession and possibly submitting new batches.
Notes
If the next batch is ready, it will be processed immediately and no new batches are submitted.
New batches are submitted only while waiting for the next one to complete. There will never be more batches submitted in parallel than the max_parallel_batches setting allows.
Returns: Return type: None

parameter_names
Â¶ Return the parameters to be inferred.

plot_state
(**kwargs)Â¶ Plot the current state of the algorithm.
Parameters:  axes (matplotlib.axes.Axes (optional)) –
 figure (matplotlib.figure.Figure (optional)) –
 xlim – xaxis limits
 ylim – yaxis limits
 interactive (bool (default False)) – If true, uses IPython.display to update the cell figure
 close – Close figure in the end of plotting. Used in the end of interactive mode.
Returns: Return type: None

pool
Â¶ Return the output pool of the inference.

prepare_new_batch
(batch_index)[source]Â¶ Prepare values for a new batch.
Parameters: batch_index (int) – next batch_index to be submitted Returns: batch – Keys should match to node names in the model. These values will override any default values or operations in those nodes. Return type: dict or None

sample
(n_samples, *args, **kwargs)Â¶ Sample from the approximate posterior.
See the other arguments from the set_objective method.
Parameters:  n_samples (int) – Number of samples to generate from the (approximate) posterior
 *args –
 **kwargs –
Returns: result
Return type:

seed
Â¶ Return the seed of the inference.

class
elfi.
BayesianOptimization
(model, target_name=None, bounds=None, initial_evidence=None, update_interval=10, target_model=None, acquisition_method=None, acq_noise_var=0, exploration_rate=10, batch_size=1, batches_per_acquisition=None, async=False, **kwargs)[source]Â¶ Bayesian Optimization of an unknown target function.
Initialize Bayesian optimization.
Parameters:  model (ElfiModel or NodeReference) –
 target_name (str or NodeReference) – Only needed if model is an ElfiModel
 bounds (dict, optional) – The region where to estimate the posterior for each parameter in model.parameters: dict(‘parameter_name’:(lower, upper), ... )`. Not used if custom target_model is given.
 initial_evidence (int, dict, optional) – Number of initial evidence or a precomputed batch dict containing parameter and discrepancy values. Default value depends on the dimensionality.
 update_interval (int, optional) – How often to update the GP hyperparameters of the target_model
 target_model (GPyRegression, optional) –
 acquisition_method (Acquisition, optional) – Method of acquiring evidence points. Defaults to LCBSC.
 acq_noise_var (float or np.array, optional) – Variance(s) of the noise added in the default LCBSC acquisition method. If an array, should be 1d specifying the variance for each dimension.
 exploration_rate (float, optional) – Exploration rate of the acquisition method
 batch_size (int, optional) – Elfi batch size. Defaults to 1.
 batches_per_acquisition (int, optional) – How many batches will be requested from the acquisition function at one go. Defaults to max_parallel_batches.
 async (bool, optional) – Allow acquisitions to be made asynchronously, i.e. do not wait for all the results from the previous acquisition before making the next. This can be more efficient with a large amount of workers (e.g. in cluster environments) but forgoes the guarantee for the exactly same result with the same initial conditions (e.g. the seed). Default False.
 **kwargs –

acq_batch_size
Â¶ Return the total number of acquisition per iteration.

batch_size
Â¶ Return the current batch_size.

extract_result
()[source]Â¶ Extract the result from the current state.
Returns: Return type: OptimizationResult

infer
(*args, vis=None, **kwargs)Â¶ Set the objective and start the iterate loop until the inference is finished.
See the other arguments from the set_objective method.
Returns: result Return type: Sample

iterate
()Â¶ Advance the inference by one iteration.
This is a way to manually progress the inference. One iteration consists of waiting and processing the result of the next batch in succession and possibly submitting new batches.
Notes
If the next batch is ready, it will be processed immediately and no new batches are submitted.
New batches are submitted only while waiting for the next one to complete. There will never be more batches submitted in parallel than the max_parallel_batches setting allows.
Returns: Return type: None

n_evidence
Â¶ Return the number of acquired evidence points.

parameter_names
Â¶ Return the parameters to be inferred.

plot_discrepancy
(axes=None, **kwargs)[source]Â¶ Plot acquired parameters vs. resulting discrepancy.
TODO: refactor

plot_state
(**options)[source]Â¶ Plot the GP surface.
This feature is still experimental and currently supports only 2D cases.

pool
Â¶ Return the output pool of the inference.

prepare_new_batch
(batch_index)[source]Â¶ Prepare values for a new batch.
Parameters: batch_index (int) – next batch_index to be submitted Returns: batch – Keys should match to node names in the model. These values will override any default values or operations in those nodes. Return type: dict or None

seed
Â¶ Return the seed of the inference.

class
elfi.
BOLFI
(model, target_name=None, bounds=None, initial_evidence=None, update_interval=10, target_model=None, acquisition_method=None, acq_noise_var=0, exploration_rate=10, batch_size=1, batches_per_acquisition=None, async=False, **kwargs)[source]Â¶ Bayesian Optimization for LikelihoodFree Inference (BOLFI).
Approximates the discrepancy function by a stochastic regression model. Discrepancy model is fit by sampling the discrepancy function at points decided by the acquisition function.
The method implements the framework introduced in Gutmann & Corander, 2016.
References
Gutmann M U, Corander J (2016). Bayesian Optimization for LikelihoodFree Inference of SimulatorBased Statistical Models. JMLR 17(125):1âˆ’47, 2016. http://jmlr.org/papers/v17/15017.html
Initialize Bayesian optimization.
Parameters:  model (ElfiModel or NodeReference) –
 target_name (str or NodeReference) – Only needed if model is an ElfiModel
 bounds (dict, optional) – The region where to estimate the posterior for each parameter in model.parameters: dict(‘parameter_name’:(lower, upper), ... )`. Not used if custom target_model is given.
 initial_evidence (int, dict, optional) – Number of initial evidence or a precomputed batch dict containing parameter and discrepancy values. Default value depends on the dimensionality.
 update_interval (int, optional) – How often to update the GP hyperparameters of the target_model
 target_model (GPyRegression, optional) –
 acquisition_method (Acquisition, optional) – Method of acquiring evidence points. Defaults to LCBSC.
 acq_noise_var (float or np.array, optional) – Variance(s) of the noise added in the default LCBSC acquisition method. If an array, should be 1d specifying the variance for each dimension.
 exploration_rate (float, optional) – Exploration rate of the acquisition method
 batch_size (int, optional) – Elfi batch size. Defaults to 1.
 batches_per_acquisition (int, optional) – How many batches will be requested from the acquisition function at one go. Defaults to max_parallel_batches.
 async (bool, optional) – Allow acquisitions to be made asynchronously, i.e. do not wait for all the results from the previous acquisition before making the next. This can be more efficient with a large amount of workers (e.g. in cluster environments) but forgoes the guarantee for the exactly same result with the same initial conditions (e.g. the seed). Default False.
 **kwargs –

acq_batch_size
Â¶ Return the total number of acquisition per iteration.

batch_size
Â¶ Return the current batch_size.

extract_posterior
(threshold=None)[source]Â¶ Return an object representing the approximate posterior.
The approximation is based on surrogate model regression.
Parameters: threshold (float, optional) – Discrepancy threshold for creating the posterior (log with log discrepancy). Returns: posterior Return type: elfi.methods.posteriors.BolfiPosterior

extract_result
()Â¶ Extract the result from the current state.
Returns: Return type: OptimizationResult

fit
(n_evidence, threshold=None)[source]Â¶ Fit the surrogate model.
Generates a regression model for the discrepancy given the parameters.
Currently only Gaussian processes are supported as surrogate models.
Parameters: threshold (float, optional) – Discrepancy threshold for creating the posterior (log with log discrepancy).

infer
(*args, vis=None, **kwargs)Â¶ Set the objective and start the iterate loop until the inference is finished.
See the other arguments from the set_objective method.
Returns: result Return type: Sample

iterate
()Â¶ Advance the inference by one iteration.
This is a way to manually progress the inference. One iteration consists of waiting and processing the result of the next batch in succession and possibly submitting new batches.
Notes
If the next batch is ready, it will be processed immediately and no new batches are submitted.
New batches are submitted only while waiting for the next one to complete. There will never be more batches submitted in parallel than the max_parallel_batches setting allows.
Returns: Return type: None

n_evidence
Â¶ Return the number of acquired evidence points.

parameter_names
Â¶ Return the parameters to be inferred.

plot_discrepancy
(axes=None, **kwargs)Â¶ Plot acquired parameters vs. resulting discrepancy.
TODO: refactor

plot_state
(**options)Â¶ Plot the GP surface.
This feature is still experimental and currently supports only 2D cases.

pool
Â¶ Return the output pool of the inference.

prepare_new_batch
(batch_index)Â¶ Prepare values for a new batch.
Parameters: batch_index (int) – next batch_index to be submitted Returns: batch – Keys should match to node names in the model. These values will override any default values or operations in those nodes. Return type: dict or None

sample
(n_samples, warmup=None, n_chains=4, threshold=None, initials=None, algorithm='nuts', n_evidence=None, **kwargs)[source]Â¶ Sample the posterior distribution of BOLFI.
Here the likelihood is defined through the cumulative density function of the standard normal distribution:
L(theta) propto F((hmu(theta)) / sigma(theta))
where h is the threshold, and mu(theta) and sigma(theta) are the posterior mean and (noisy) standard deviation of the associated Gaussian process.
The sampling is performed with an MCMC sampler (the NoUTurn Sampler, NUTS).
Parameters:  n_samples (int) – Number of requested samples from the posterior for each chain. This includes warmup, and note that the effective sample size is usually considerably smaller.
 warmpup (int, optional) – Length of warmup sequence in MCMC sampling. Defaults to n_samples//2.
 n_chains (int, optional) – Number of independent chains.
 threshold (float, optional) – The threshold (bandwidth) for posterior (give as log if log discrepancy).
 initials (np.array of shape (n_chains, n_params), optional) – Initial values for the sampled parameters for each chain. Defaults to best evidence points.
 algorithm (string, optional) – Sampling algorithm to use. Currently only ‘nuts’ is supported.
 n_evidence (int) – If the regression model is not fitted yet, specify the amount of evidence
Returns: Return type:

seed
Â¶ Return the seed of the inference.

set_objective
(n_evidence=None)Â¶ Set objective for inference.
You can continue BO by giving a larger n_evidence.
Parameters: n_evidence (int) – Number of total evidence for the GP fitting. This includes any initial evidence.

update
(batch, batch_index)Â¶ Update the GP regression model of the target node with a new batch.
Parameters:  batch (dict) – dict with self.outputs as keys and the corresponding outputs for the batch as values
 batch_index (int) –
Result objects

class
elfi.methods.results.
OptimizationResult
(x_min, **kwargs)[source]Â¶ Base class for results from optimization.
Initialize result.
Parameters:  x_min – The optimized parameters
 **kwargs – See ParameterInferenceResult

class
elfi.methods.results.
Sample
(method_name, outputs, parameter_names, discrepancy_name=None, weights=None, **kwargs)[source]Â¶ Sampling results from inference methods.
Initialize result.
Parameters:  method_name (string) – Name of inference method.
 outputs (dict) – Dictionary with outputs from the nodes, e.g. samples.
 parameter_names (list) – Names of the parameter nodes
 discrepancy_name (string, optional) – Name of the discrepancy in outputs.
 weights (array_like) –
 **kwargs – Other meta information for the result

dim
Â¶ Return the number of parameters.

discrepancies
Â¶ Return the discrepancy values.

n_samples
Â¶ Return the number of samples.

plot_marginals
(selector=None, bins=20, axes=None, **kwargs)[source]Â¶ Plot marginal distributions for parameters.
Parameters:  selector (iterable of ints or strings, optional) – Indices or keys to use from samples. Default to all.
 bins (int, optional) – Number of bins in histograms.
 axes (one or an iterable of plt.Axes, optional) –
Returns: axes
Return type: np.array of plt.Axes

plot_pairs
(selector=None, bins=20, axes=None, **kwargs)[source]Â¶ Plot pairwise relationships as a matrix with marginals on the diagonal.
The yaxis of marginal histograms are scaled.
Parameters:  selector (iterable of ints or strings, optional) – Indices or keys to use from samples. Default to all.
 bins (int, optional) – Number of bins in histograms.
 axes (one or an iterable of plt.Axes, optional) –
Returns: axes
Return type: np.array of plt.Axes

sample_means
Â¶ Evaluate weighted averages of sampled parameters.
Returns: Return type: OrderedDict

sample_means_array
Â¶ Evaluate weighted averages of sampled parameters.
Returns: Return type: np.array

samples_array
Â¶ Return the samples as an array.
The columns are in the same order as in self.parameter_names.
Returns: Return type: list of np.arrays

class
elfi.methods.results.
SmcSample
(method_name, outputs, parameter_names, populations, *args, **kwargs)[source]Â¶ Container for results from SMCABC.
Initialize result.
Parameters:  method_name (str) –
 outputs (dict) –
 parameter_names (list) –
 populations (list[Sample]) – List of Sample objects
 args –
 kwargs –

dim
Â¶ Return the number of parameters.

discrepancies
Â¶ Return the discrepancy values.

n_populations
Â¶ Return the number of populations.

n_samples
Â¶ Return the number of samples.

plot_marginals
(selector=None, bins=20, axes=None, all=False, **kwargs)[source]Â¶ Plot marginal distributions for parameters for all populations.
Parameters:  selector (iterable of ints or strings, optional) – Indices or keys to use from samples. Default to all.
 bins (int, optional) – Number of bins in histograms.
 axes (one or an iterable of plt.Axes, optional) –
 all (bool, optional) – Plot the marginals of all populations

plot_pairs
(selector=None, bins=20, axes=None, all=False, **kwargs)[source]Â¶ Plot pairwise relationships as a matrix with marginals on the diagonal.
The yaxis of marginal histograms are scaled.
Parameters:  selector (iterable of ints or strings, optional) – Indices or keys to use from samples. Default to all.
 bins (int, optional) – Number of bins in histograms.
 axes (one or an iterable of plt.Axes, optional) –
 all (bool, optional) – Plot for all populations

sample_means
Â¶ Evaluate weighted averages of sampled parameters.
Returns: Return type: OrderedDict

sample_means_array
Â¶ Evaluate weighted averages of sampled parameters.
Returns: Return type: np.array

sample_means_summary
(all=False)[source]Â¶ Print a representation of sample means.
Parameters: all (bool, optional) – Whether to print the means for all populations separately, or just the final population (default).

samples_array
Â¶ Return the samples as an array.
The columns are in the same order as in self.parameter_names.
Returns: Return type: list of np.arrays

class
elfi.methods.results.
BolfiSample
(method_name, chains, parameter_names, warmup, **kwargs)[source]Â¶ Container for results from BOLFI.
Initialize result.
Parameters:  method_name (string) – Name of inference method.
 chains (np.array) – Chains from sampling, warmup included. Shape: (n_chains, n_samples, n_parameters).
 parameter_names (list : list of strings) – List of names in the outputs dict that refer to model parameters.
 warmup (int) – Number of warmup iterations in chains.

dim
Â¶ Return the number of parameters.

discrepancies
Â¶ Return the discrepancy values.

n_samples
Â¶ Return the number of samples.

plot_marginals
(selector=None, bins=20, axes=None, **kwargs)Â¶ Plot marginal distributions for parameters.
Parameters:  selector (iterable of ints or strings, optional) – Indices or keys to use from samples. Default to all.
 bins (int, optional) – Number of bins in histograms.
 axes (one or an iterable of plt.Axes, optional) –
Returns: axes
Return type: np.array of plt.Axes

plot_pairs
(selector=None, bins=20, axes=None, **kwargs)Â¶ Plot pairwise relationships as a matrix with marginals on the diagonal.
The yaxis of marginal histograms are scaled.
Parameters:  selector (iterable of ints or strings, optional) – Indices or keys to use from samples. Default to all.
 bins (int, optional) – Number of bins in histograms.
 axes (one or an iterable of plt.Axes, optional) –
Returns: axes
Return type: np.array of plt.Axes

sample_means
Â¶ Evaluate weighted averages of sampled parameters.
Returns: Return type: OrderedDict

sample_means_array
Â¶ Evaluate weighted averages of sampled parameters.
Returns: Return type: np.array

sample_means_summary
()Â¶ Print a representation of sample means.

samples_array
Â¶ Return the samples as an array.
The columns are in the same order as in self.parameter_names.
Returns: Return type: list of np.arrays

summary
()Â¶ Print a verbose summary of contained results.
Postprocessing

elfi.
adjust_posterior
(sample, model, summary_names, parameter_names=None, adjustment='linear')[source]Â¶ Adjust the posterior using local regression.
Note that the summary nodes need to be explicitly included to the sample object with the output_names keyword argument when performing the inference.
Parameters:  sample (elfi.methods.results.Sample) – a sample object from an ABC algorithm
 model (elfi.ElfiModel) – the inference model
 summary_names (list[str]) – names of the summary nodes
 parameter_names (list[str] (optional)) – names of the parameters
 adjustment (RegressionAdjustment or string) –
a regression adjustment object or a string specification
 Accepted values for the string specification:
 ‘linear’
Returns: a Sample object with the adjusted posterior
Return type: Examples
>>> import elfi >>> from elfi.examples import gauss >>> m = gauss.get_model() >>> res = elfi.Rejection(m['d'], output_names=['ss_mean', 'ss_var']).sample(1000) >>> adj = adjust_posterior(res, m, ['ss_mean', 'ss_var'], ['mu'], LinearAdjustment())

class
elfi.methods.post_processing.
LinearAdjustment
(**kwargs)[source]Â¶ Regression adjustment using a local linear model.

adjust
()Â¶ Adjust the posterior.
Only the nonfinite values used to fit the regression model will be adjusted.
Returns: Return type: a Sample object containing the adjusted posterior

fit
(sample, model, summary_names, parameter_names=None)Â¶ Fit a regression adjustment model to the posterior sample.
Nonfinite values in the summary statistics and parameters will be omitted.
Parameters:  sample (elfi.methods.Sample) – a sample object from an ABC method
 model (elfi.ElfiModel) – the inference model
 summary_names (list[str]) – a list of names for the summary nodes
 parameter_names (list[str] (optional)) – a list of parameter names

Diagnostics

class
elfi.
TwoStageSelection
(simulator, fn_distance, list_ss=None, prepared_ss=None, max_cardinality=4, seed=0)[source]Â¶ Perform the summarystatistics selection proposed by Nunes and Balding (2010).
The user can provide a list of summary statistics as list_ss, and let ELFI to combine them, or provide some already combined summary statistics as prepared_ss.
The rationale of the Two Stage procedure procedure is the following:
 First, the module computes or accepts the combinations of the candidate summary statistics.
 In Stage 1, each summarystatistics combination is evaluated using the Minimum Entropy algorithm.
 In Stage 2, the minimumentropy combination is selected, and the ‘closest’ datasets are identified.
 Further in Stage 2, for each summarystatistics combination, the mean root sum of squared errors (MRSSE) is calculated over all ‘closest datasets’, and the minimumMRSSE combination is chosen as the one with the optimal performance.
References
[1] Nunes, M. A., & Balding, D. J. (2010). On optimal selection of summary statistics for approximate Bayesian computation. Statistical applications in genetics and molecular biology, 9(1). [2] Blum, M. G., Nunes, M. A., Prangle, D., & Sisson, S. A. (2013). A comparative review of dimension reduction methods in approximate Bayesian computation. Statistical Science, 28(2), 189208.
Initialise the summarystatistics selection for the Two Stage Procedure.
Parameters:  simulator (elfi.Node) – Node (often elfi.Simulator) for which the summary statistics will be applied. The node is the final node of a coherent ElfiModel (i.e. it has no child nodes).
 fn_distance (str or callable function) – Distance metric, consult the elfi.Distance documentation for calling as a string.
 list_ss (List of callable functions, optional) – List of candidate summary statistics.
 prepared_ss (List of lists of callable functions, optional) – List of prepared combinations of candidate summary statistics. No other combinations will be evaluated.
 max_cardinality (int, optional) – Maximum cardinality of a candidate summarystatistics combination.
 seed (int, optional) –

run
(n_sim, n_acc=None, n_closest=None, batch_size=1, k=4)[source]Â¶ Run the Two Stage Procedure for identifying relevant summary statistics.
Parameters:  n_sim (int) – Number of the total ABCrejection simulations.
 n_acc (int, optional) – Number of the accepted ABCrejection simulations.
 n_closest (int, optional) – Number of the ‘closest’ datasets (i.e., the closest n simulation datasets w.r.t the observations).
 batch_size (int, optional) – Number of samples per batch.
 k (int, optional) – Parameter for the kthnearestneighbour search performed in the minimumentropy step (in Nunes & Balding, 2010 it is fixed to 4).
Returns: Summarystatistics combination showing the optimal performance.
Return type: array_like
Acquisition methods

class
elfi.methods.bo.acquisition.
LCBSC
(*args, delta=None, **kwargs)[source]Â¶ Lower Confidence Bound Selection Criterion.
Srinivas et al. call this GPLCB.
LCBSC uses the parameter delta which is here equivalent to 1/exploration_rate.
Parameter delta should be in (0, 1) for the theoretical results to hold. The theoretical upper bound for total regret in Srinivas et al. has a probability greater or equal to 1  delta, so values of delta very close to 1 or over it do not make much sense in that respect.
Delta is roughly the exploitation tendency of the acquisition function.
References
N. Srinivas, A. Krause, S. M. Kakade, and M. Seeger. Gaussian process optimization in the bandit setting: No regret and experimental design. In Proc. International Conference on Machine Learning (ICML), 2010
E. Brochu, V.M. Cora, and N. de Freitas. A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. arXiv:1012.2599, 2010.
Notes
The formula presented in Brochu (pp. 15) seems to be from Srinivas et al. Theorem 2. However, instead of having t**(d/2 + 2) in beta_t, it seems that the correct form would be t**(2d + 2).
Initialize LCBSC.
Parameters:  args –
 delta (float, optional) – In between (0, 1). Default is 1/exploration_rate. If given, overrides the exploration_rate.
 kwargs –

acquire
(n, t=None)Â¶ Return the next batch of acquisition points.
Gaussian noise ~N(0, self.noise_var) is added to the acquired points.
Parameters:  n (int) – Number of acquisition points to return.
 t (int) – Current acq_batch_index (starting from 0).
Returns: x – The shape is (n, input_dim)
Return type: np.ndarray

delta
Â¶ Return the inverse of exploration rate.

class
elfi.methods.bo.acquisition.
MaxVar
(quantile_eps=0.01, *args, **opts)[source]Â¶ The maximum variance acquisition method.
The next evaluation point is acquired in the maximiser of the variance of the unnormalised approximate posterior.
theta_{t+1} = arg max Var(p(theta) * p_a(theta)),
where the unnormalised likelihood p_a is defined using the CDF of normal distribution, Phi, as follows:
 p_a(theta) =
 (Phi((epsilon  mu_{1:t}(theta)) / sqrt(v_{1:t}(theta) + sigma2_n))),
where epsilon is the ABC threshold, mu_{1:t} and v_{1:t} are determined by the Gaussian process, sigma2_n is the noise.
References
[1] JÃ¤rvenpÃ¤Ã¤ et al. (2017). arXiv:1704.00520 [2] Gutmann M U, Corander J (2016). Bayesian Optimization for LikelihoodFree Inference of SimulatorBased Statistical Models. JMLR 17(125):1âˆ’47, 2016. http://jmlr.org/papers/v17/15017.html
Initialise MaxVar.
Parameters: quantile_eps (int, optional) – Quantile of the observed discrepancies used in setting the ABC threshold. 
acquire
(n, t=None)[source]Â¶ Acquire a batch of acquisition points.
Parameters:  n (int) – Number of acquisitions.
 t (int, optional) – Current iteration, (unused).
Returns: Coordinates of the yielded acquisition points.
Return type: array_like

evaluate
(theta_new, t=None)[source]Â¶ Evaluate the acquisition function at the location theta_new.
Parameters:  theta_new (array_like) – Evaluation coordinates.
 t (int, optional) – Current iteration, (unused).
Returns: Variance of the approximate posterior.
Return type: array_like

evaluate_gradient
(theta_new, t=None)[source]Â¶ Evaluate the acquisition function’s gradient at the location theta_new.
Parameters:  theta_new (array_like) – Evaluation coordinates.
 t (int, optional) – Current iteration, (unused).
Returns: Gradient of the variance of the approximate posterior
Return type: array_like

class
elfi.methods.bo.acquisition.
RandMaxVar
(quantile_eps=0.01, sampler='nuts', n_samples=50, limit_faulty_init=10, sigma_proposals_metropolis=None, *args, **opts)[source]Â¶ The randomised maximum variance acquisition method.
The next evaluation point is sampled from the density corresponding to the variance of the unnormalised approximate posterior (The MaxVar acquisition function).
theta_{t+1} ~ q(theta),
where q(theta) propto Var(p(theta) * p_a(theta)) and the unnormalised likelihood p_a is defined using the CDF of normal distribution, Phi, as follows:
 p_a(theta) =
 (Phi((epsilon  mu_{1:t}(theta)) / sqrt(v_{1:t}(theta) + sigma2_n))),
where epsilon is the ABC threshold, mu_{1:t} and v_{1:t} are determined by the Gaussian process, sigma2_n is the noise.
References
[1] arXiv:1704.00520 (JÃ¤rvenpÃ¤Ã¤ et al., 2017)
Initialise RandMaxVar.
Parameters:  quantile_eps (int, optional) – Quantile of the observed discrepancies used in setting the ABC threshold.
 sampler (string, optional) – Name of the sampler (options: metropolis, nuts).
 n_samples (int, optional) – Length of the sampler’s chain for obtaining the acquisitions.
 limit_faulty_init (int, optional) – Limit for the iterations used to obtain the sampler’s initial points.
 sigma_proposals_metropolis (array_like, optional) – Standard deviation proposals for tuning the metropolis sampler. For the default settings, the sigmas are set to the 1/10 of the parameter intervals’ length.

acquire
(n, t=None)[source]Â¶ Acquire a batch of acquisition points.
Parameters:  n (int) – Number of acquisitions.
 t (int, optional) – Current iteration, (unused).
Returns: Coordinates of the yielded acquisition points.
Return type: array_like

evaluate
(theta_new, t=None)Â¶ Evaluate the acquisition function at the location theta_new.
Parameters:  theta_new (array_like) – Evaluation coordinates.
 t (int, optional) – Current iteration, (unused).
Returns: Variance of the approximate posterior.
Return type: array_like

evaluate_gradient
(theta_new, t=None)Â¶ Evaluate the acquisition function’s gradient at the location theta_new.
Parameters:  theta_new (array_like) – Evaluation coordinates.
 t (int, optional) – Current iteration, (unused).
Returns: Gradient of the variance of the approximate posterior
Return type: array_like

class
elfi.methods.bo.acquisition.
ExpIntVar
(quantile_eps=0.01, integration='grid', d_grid=0.2, n_samples_imp=100, iter_imp=2, sampler='nuts', n_samples=2000, sigma_proposals_metropolis=None, *args, **opts)[source]Â¶ The Expected Integrated Variance (ExpIntVar) acquisition method.
Essentially, we define a loss function that measures the overall uncertainty in the unnormalised ABC posterior over the parameter space. The value of the loss function depends on the next simulation and thus the next evaluation location theta^* is chosen to minimise the expected loss.
theta_{t+1} = arg min_{theta^* in Theta} L_{1:t}(theta^*), where
Theta is the parameter space, and L is the expected loss function approximated as follows:
 L_{1:t}(theta^*) approx 2 * sum_{i=1}^s (omega^i * p^2(theta^i)
 w_{1:t+1})(theta^i, theta^*), where
omega^i is an importance weight, p^2(theta^i) is the prior squared, and w_{1:t+1})(theta^i, theta^*) is the expected variance of the unnormalised ABC posterior at theta^i after running the simulation model with parameter theta^*
References
[1] arXiv:1704.00520 (JÃ¤rvenpÃ¤Ã¤ et al., 2017)
Initialise ExpIntVar.
Parameters:  quantile_eps (int, optional) – Quantile of the observed discrepancies used in setting the discrepancy threshold.
 integration (str, optional) – Integration method. Options:  grid (points are taken uniformly): more accurate yet computationally expensive in high dimensions;  importance (points are taken based on the importance weight): less accurate though applicable in high dimensions.
 d_grid (float, optional) – Grid tightness.
 n_samples_imp (int, optional) – Number of importance samples.
 iter_imp (int, optional) – Gap between acquisition iterations in performing importance sampling.
 sampler (string, optional) – Sampler for generating random numbers from the proposal distribution for IS. (Options: metropolis, nuts.)
 n_samples (int, optional) – Chain length for the sampler that generates the random numbers from the proposal distribution for IS.
 sigma_proposals_metropolis (array_like, optional) – Standard deviation proposals for tuning the metropolis sampler.

acquire
(n, t)[source]Â¶ Acquire a batch of acquisition points.
Parameters:  n (int) – Number of acquisitions.
 t (int) – Current iteration.
Returns: Coordinates of the yielded acquisition points.
Return type: array_like

evaluate
(theta_new, t=None)[source]Â¶ Evaluate the acquisition function at the location theta_new.
Parameters:  theta_new (array_like) – Evaluation coordinates.
 t (int, optional) – Current iteration, (unused).
Returns: Expected loss’s term dependent on theta_new.
Return type: array_like

evaluate_gradient
(theta_new, t=None)Â¶ Evaluate the acquisition function’s gradient at the location theta_new.
Parameters:  theta_new (array_like) – Evaluation coordinates.
 t (int, optional) – Current iteration, (unused).
Returns: Gradient of the variance of the approximate posterior
Return type: array_like

class
elfi.methods.bo.acquisition.
UniformAcquisition
(model, prior=None, n_inits=10, max_opt_iters=1000, noise_var=None, exploration_rate=10, seed=None)[source]Â¶ Acquisition from uniform distribution.
Initialize AcquisitionBase.
Parameters:  model (an object with attributes) –
 input_dim : int
 bounds : tuple of length ‘input_dim’ of tuples (min, max)
 and methods
 evaluate(x) : function that returns model (mean, var, std)
 prior (scipylike distribution, optional) – By default uniform distribution within model bounds.
 n_inits (int, optional) – Number of initialization points in internal optimization.
 max_opt_iters (int, optional) – Max iterations to optimize when finding the next point.
 noise_var (float or np.array, optional) – Acquisition noise variance for adding noise to the points near the optimized location. If array, must be 1d specifying the variance for different dimensions. Default: no added noise.
 exploration_rate (float, optional) – Exploration rate of the acquisition function (if supported)
 seed (int, optional) – Seed for getting consistent acquisition results. Used in getting random starting locations in acquisition function optimization.

acquire
(n, t=None)[source]Â¶ Return random points from uniform distribution.
Parameters:  n (int) – Number of acquisition points to return.
 t (int, optional) – (unused)
Returns: x – The shape is (n, input_dim)
Return type: np.ndarray

evaluate
(x, t=None)Â¶ Evaluate the acquisition function at ‘x’.
Parameters:  x (numpy.array) –
 t (int) – current iteration (starting from 0)

evaluate_gradient
(x, t=None)Â¶ Evaluate the gradient of acquisition function at ‘x’.
Parameters:  x (numpy.array) –
 t (int) – Current iteration (starting from 0).
 model (an object with attributes) –
OtherÂ¶
Data pools

class
elfi.
OutputPool
(outputs=None, name=None, prefix=None)[source]Â¶ Store node outputs to dictionarylike stores.
The default store is a Python dictionary.
Notes
Saving the store requires that all the stores are pickleable.
Arbitrary objects that support simple array indexing can be used as stores by using the elfi.store.ArrayObjectStore class.
See the elfi.store.StoreBase interfaces if you wish to implement your own ELFI compatible store. Basically any object that fulfills the Pythons dictionary api will work as a store in the pool.
Initialize OutputPool.
Depending on the algorithm, some of these values may be reused after making some changes to ElfiModel thus speeding up the inference significantly. For instance, if all the simulations are stored in Rejection sampling, one can change the summaries and distances without having to rerun the simulator.
Parameters:  outputs (list, dict, optional) – List of node names which to store or a dictionary with existing stores. The stores are created on demand.
 name (str, optional) – Name of the pool. Used to open a saved pool from disk.
 prefix (str, optional) – Path to directory under which elfi.ArrayPool will place its folder. Default is a relative path ./pools.
Returns: instance
Return type: 
add_store
(node, store=None)[source]Â¶ Add a store object for the node.
Parameters:  node (str) –
 store (dict, StoreBase, optional) –

close
()[source]Â¶ Save and close the stores that support it.
The pool will not be usable afterwards.

flush
()[source]Â¶ Flush all data from the stores.
If the store does not support flushing, do nothing.

get_batch
(batch_index, output_names=None)[source]Â¶ Return a batch from the stores of the pool.
Parameters:  batch_index (int) –
 output_names (list) – which outputs to include to the batch
Returns: batch
Return type: dict

has_context
Â¶ Check if current pool has context information.

classmethod
open
(name, prefix=None)[source]Â¶ Open a closed or saved ArrayPool from disk.
Parameters:  name (str) –
 prefix (str, optional) –
Returns: Return type:

output_names
Â¶ Return a list of stored names.

path
Â¶ Return the path to the pool.

class
elfi.
ArrayPool
(outputs=None, name=None, prefix=None)[source]Â¶ OutputPool that uses binary .npy files as default stores.
The default store medium for output data is a NumPy binary .npy file for NumPy array data. You can however also add other types of stores as well.
Notes
The default store is implemented in elfi.store.NpyStore that uses NpyArrays as stores. The NpyArray is a wrapper over NumPy .npy binary file for array data and supports appending the .npy file. It uses the .npy format 2.0 files.
Initialize OutputPool.
Depending on the algorithm, some of these values may be reused after making some changes to ElfiModel thus speeding up the inference significantly. For instance, if all the simulations are stored in Rejection sampling, one can change the summaries and distances without having to rerun the simulator.
Parameters:  outputs (list, dict, optional) – List of node names which to store or a dictionary with existing stores. The stores are created on demand.
 name (str, optional) – Name of the pool. Used to open a saved pool from disk.
 prefix (str, optional) – Path to directory under which elfi.ArrayPool will place its folder. Default is a relative path ./pools.
Returns: instance
Return type: 
add_batch
(batch, batch_index)Â¶ Add the outputs from the batch to their stores.

add_store
(node, store=None)Â¶ Add a store object for the node.
Parameters:  node (str) –
 store (dict, StoreBase, optional) –

clear
()Â¶ Remove all data from the stores.

close
()Â¶ Save and close the stores that support it.
The pool will not be usable afterwards.

delete
()Â¶ Remove all persisted data from disk.

flush
()Â¶ Flush all data from the stores.
If the store does not support flushing, do nothing.

get_batch
(batch_index, output_names=None)Â¶ Return a batch from the stores of the pool.
Parameters:  batch_index (int) –
 output_names (list) – which outputs to include to the batch
Returns: batch
Return type: dict

get_store
(node)Â¶ Return the store for node.

has_context
Â¶ Check if current pool has context information.

has_store
(node)Â¶ Check if node is in stores.

open
(name, prefix=None)Â¶ Open a closed or saved ArrayPool from disk.
Parameters:  name (str) –
 prefix (str, optional) –
Returns: Return type:

output_names
Â¶ Return a list of stored names.

path
Â¶ Return the path to the pool.

remove_batch
(batch_index)Â¶ Remove the batch from all stores.

remove_store
(node)Â¶ Remove and return a store from the pool.
Parameters: node (str) – Returns: The removed store Return type: store

save
()Â¶ Save the pool to disk.
This will use pickle to store the pool under self.path.

set_context
(context)Â¶ Set the context of the pool.
The pool needs to know the batch_size and the seed.
Notes
Also sets the name of the pool if not set already.
Parameters: context (elfi.ComputationContext) –
Module functions
Tools

tools.
vectorize
(operation, constants=None, dtype=None)Â¶ Vectorize an operation.
Helper for cases when you have an operation that does not support vector arguments. This tool is still experimental and may not work in all cases.
Parameters:  operation (callable) – Operation to vectorize.
 constants (tuple, list, optional) – A mask for constants in inputs, e.g. (0, 2) would indicate that the first and third positional inputs are constants. The constants will be passed as they are to each operation call.
 dtype (np.dtype, bool[False], optional) – If None, numpy converts a list of outputs automatically. In some cases this produces non desired results. If you wish to keep the outputs as they are with no conversion, specify dtype=False. This results into a 1d object numpy array with outputs as they were returned.
Notes
This is a convenience method that uses a for loop internally for the vectorization. For best performance, one should aim to implement vectorized operations (by using e.g. numpy functions that are mostly vectorized) if at all possible.
Examples
# This form works in most cases vectorized_simulator = elfi.tools.vectorize(simulator) # Tell that the second and third argument to the simulator will be a constant vectorized_simulator = elfi.tools.vectorize(simulator, [1, 2]) elfi.Simulator(vectorized_simulator, prior, constant_1, constant_2) # Tell the vectorizer that it should not do any conversion to the outputs vectorized_simulator = elfi.tools.vectorize(simulator, dtype=False)

tools.
external_operation
(command, process_result=None, prepare_inputs=None, sep=' ', stdout=True, subprocess_kwargs=None)Â¶ Wrap an external command as a Python callable (function).
The external command can be e.g. a shell script, or an executable file.
Parameters:  command (str) – Command to execute. Arguments can be passed to the executable by using Python’s format strings, e.g. “myscript.sh {0} {batch_size} –seed {seed}”. The command is expected to write to stdout. Since random_state is python specific object, a seed keyword argument will be available to operations that use random_state.
 process_result (callable, np.dtype, str, optional) – Callable result handler with a signature output = callable(result, *inputs, **kwinputs). Here the result is either the stdout or subprocess.CompletedProcess depending on the stdout flag below. The inputs and kwinputs will come from ELFI. The default handler converts the stdout to numpy array with array = np.fromstring(stdout, sep=sep). If process_result is np.dtype or a string, then the stdout data is casted to that type with stdout = np.fromstring(stdout, sep=sep, dtype=process_result).
 prepare_inputs (callable, optional) – Callable with a signature inputs, kwinputs = callable(*inputs, **kwinputs). The inputs will come from elfi.
 sep (str, optional) – Separator to use with the default process_result handler. Default is a space ‘ ‘. If you specify your own callable to process_result this value has no effect.
 stdout (bool, optional) – Pass the process_result handler the stdout instead of the subprocess.CompletedProcess instance. Default is true.
 subprocess_kwargs (dict, optional) – Options for Python’s subprocess.run that is used to run the external command. Defaults are shell=True, check=True. See the subprocess documentation for more details.
Examples
>>> import elfi >>> op = elfi.tools.external_operation('echo 1 {0}', process_result='int8') >>> >>> constant = elfi.Constant(123) >>> simulator = elfi.Simulator(op, constant) >>> simulator.generate() array([ 1, 123], dtype=int8)
Returns: operation – ELFI compatible operation that can be used e.g. as a simulator. Return type: callable
Frequently Asked QuestionsÂ¶
Below are answers to some common questions asked about ELFI.
Q: My uniform prior elfi.Prior('uniform', 1, 2)
does not seem to be right as it
produces outputs from the interval (1, 3).
A: The distributions defined by strings are those from scipy.stats
and follow
their definitions. There the uniform distribution uses the location/scale definition, so
the first argument defines the starting point of the interval and the second its length.
This tutorial is generated from a Jupyter notebook that can be found here.
ELFI tutorialÂ¶
This tutorial covers the basics of using ELFI, i.e. how to make models, save results for later use and run different inference algorithms.
Let’s begin by importing libraries that we will use and specify some settings.
import time
import numpy as np
import scipy.stats
import matplotlib
import matplotlib.pyplot as plt
import logging
logging.basicConfig(level=logging.INFO)
%matplotlib inline
%precision 2
# Set an arbitrary seed and a global random state to keep the randomly generated quantities the same between runs
seed = 20170530
np.random.seed(seed)
Inference with ELFI: case MA(2) modelÂ¶
Throughout this tutorial we will use the 2nd order moving average model MA(2) as an example. MA(2) is a common model used in univariate time series analysis. Assuming zero mean it can be written as
where \(\theta_1, \theta_2 \in \mathbb{R}\) and \((w_k)_{k\in \mathbb{Z}} \sim N(0,1)\) represents an independent and identically distributed sequence of white noise.
The observed data and the inference problemÂ¶
In this tutorial, our task is to infer the parameters \(\theta_1, \theta_2\) given a sequence of 100 observations \(y\) that originate from an MA(2) process. Let’s define the MA(2) simulator as a Python function:
def MA2(t1, t2, n_obs=100, batch_size=1, random_state=None):
# Make inputs 2d arrays for numpy broadcasting with w
t1 = np.asanyarray(t1).reshape((1, 1))
t2 = np.asanyarray(t2).reshape((1, 1))
random_state = random_state or np.random
w = random_state.randn(batch_size, n_obs+2) # i.i.d. sequence ~ N(0,1)
x = w[:, 2:] + t1*w[:, 1:1] + t2*w[:, :2]
return x
Above, t1
, t2
, and n_obs
are the arguments specific to the
MA2 process. The latter two, batch_size
and random_state
are
ELFI specific keyword arguments. The batch_size
argument tells how
many simulations are needed. The random_state
argument is for
generating random quantities in your simulator. It is a
numpy.RandomState
object that has all the same methods as
numpy.random
module has. It is used for ensuring consistent results
and handling random number generation in parallel settings.
VectorizationÂ¶
What is the purpose of the batch_size
argument? In ELFI, operations
are vectorized, meaning that instead of simulating a single MA2 sequence
at a time, we simulate a batch of them. A vectorized function takes
vectors as inputs, and computes the output for each element in the
vector. Vectorization is a way to make operations efficient in Python.
Above we rely on numpy to carry out the vectorized calculations.
In this case the arguments t1
and t2
are going to be vectors of
length batch_size
and the method returns a 2d array with the
simulations on the rows. Notice that for convenience, the funtion also
works with scalars that are first converted to vectors.
Note
There is a builtin tool (elfi.tools.vectorize) in ELFI to vectorize operations that are not vectorized. It is basically a for loop wrapper.
Important
In order to guarantee a consistent state of pseudorandom number generation, the simulator must have random_state as a keyword argument for reading in a numpy.RandomState object.
Let’s now use this simulator to create toy observations. We will use parameter values \(\theta_1=0.6, \theta_2=0.2\) as in *Marin et al. (2012)* and then try to infer these parameter values back based on the toy observed data alone.
# true parameters
t1_true = 0.6
t2_true = 0.2
y_obs = MA2(t1_true, t2_true)
# Plot the observed sequence
plt.figure(figsize=(11, 6));
plt.plot(y_obs.ravel());
# To illustrate the stochasticity, let's plot a couple of more observations with the same true parameters:
plt.plot(MA2(t1_true, t2_true).ravel());
plt.plot(MA2(t1_true, t2_true).ravel());
Approximate Bayesian ComputationÂ¶
Standard statistical inference methods rely on the use of the
likelihood function. Given a configuration of the parameters, the
likelihood function quantifies how likely it is that values of the
parameters produced the observed data. In our simple example case above
however, evaluating the likelihood is difficult due to the unobserved
latent sequence (variable w
in the simulator code). In many real
world applications the likelihood function is not available or it is too
expensive to evaluate preventing the use of traditional inference
methods.
One way to approach this problem is to use Approximate Bayesian Computation (ABC) which is a statistically based method replacing the use of the likelihood function with a simulator of the data. Loosely speaking, it is based on the intuition that similar data is likely to have been produced by similar parameters. Looking at the picture above, in essence we would keep simulating until we have found enough sequences that are similar to the observed sequence. Although the idea may appear inapplicable for the task at hand, you will soon see that it does work. For more information about ABC, please see e.g.
 Lintusaari, J., Gutmann, M. U., Dutta, R., Kaski, S., and Corander, J. (2016). Fundamentals and recent developments in approximate Bayesian computation. *Systematic Biology*, doi: 10.1093/sysbio/syw077.
 Marin, J.M., Pudlo, P., Robert, C. P., and Ryder, R. J. (2012). Approximate Bayesian computational methods. *Statistics and Computing*, 22(6):1167â€“1180.
 https://en.wikipedia.org/wiki/Approximate_Bayesian_computation
Defining the modelÂ¶
ELFI includes an easy to use generative modeling syntax, where the generative model is specified as a directed acyclic graph (DAG). This provides an intuitive means to describe rather complex dependencies conveniently. Often the target of the generative model is a distance between the simulated and observed data. To start creating our model, we will first import ELFI:
import elfi
As is usual in Bayesian statistical inference, we need to define prior
distributions for the unknown parameters \(\theta_1, \theta_2\). In
ELFI the priors can be any of the continuous and discrete distributions
available in scipy.stats
(for custom priors, see
below). For simplicity, let’s start by assuming
that both parameters follow Uniform(0, 2)
.
# a node is defined by giving a distribution from scipy.stats together with any arguments (here 0 and 2)
t1 = elfi.Prior(scipy.stats.uniform, 0, 2)
# ELFI also supports giving the scipy.stats distributions as strings
t2 = elfi.Prior('uniform', 0, 2)
Next, we define the simulator node with the MA2
function above,
and give the priors to it as arguments. This means that the parameters
for the simulations will be drawn from the priors. Because we have the
observed data available for this node, we provide it here as well:
Y = elfi.Simulator(MA2, t1, t2, observed=y_obs)
But how does one compare the simulated sequences with the observed sequence? Looking at the plot of just a few observed sequences above, a direct pointwise comparison would probably not work very well: the three sequences look quite different although they were generated with the same parameter values. Indeed, the comparison of simulated sequences is often the most difficult (and ad hoc) part of ABC. Typically one chooses one or more summary statistics and then calculates the discrepancy between those.
Here, we will apply the intuition arising from the definition of the MA(2) process, and use the autocovariances with lags 1 and 2 as the summary statistics:
def autocov(x, lag=1):
C = np.mean(x[:,lag:] * x[:,:lag], axis=1)
return C
As is familiar by now, a Summary
node is defined by giving the
autocovariance function and the simulated data (which includes the
observed as well):
S1 = elfi.Summary(autocov, Y)
S2 = elfi.Summary(autocov, Y, 2) # the optional keyword lag is given the value 2
Here, we choose the discrepancy as the common Euclidean L2distance.
ELFI can use many common distances directly from
scipy.spatial.distance
like this:
# Finish the model with the final node that calculates the squared distance (S1_simS1_obs)**2 + (S2_simS2_obs)**2
d = elfi.Distance('euclidean', S1, S2)
One may wish to use a distance function that is unavailable in
scipy.spatial.distance
. ELFI supports defining a custom
distance/discrepancy functions as well (see the documentation for
elfi.Distance
and elfi.Discrepancy
).
Now that the inference model is defined, ELFI can visualize the model as a DAG.
elfi.draw(d) # just give it a node in the model, or the model itself (d.model)
Note
You will need the Graphviz software as well as the graphviz Python package (https://pypi.python.org/pypi/graphviz) for drawing this. The software is already installed in many unixlike OS.
Modifying the modelÂ¶
Although the above definition is perfectly valid, let’s use the same priors as in *Marin et al. (2012)* that guarantee that the problem will be identifiable (loosely speaking, the likelihood willl have just one mode). Marin et al. used priors for which \(2<\theta_1<2\) with \(\theta_1+\theta_2>1\) and \(\theta_1\theta_2<1\) i.e. the parameters are sampled from a triangle (see below).
Custom priorsÂ¶
In ELFI, custom distributions can be defined similar to distributions in
scipy.stats
(i.e. they need to have at least the rvs
method
implemented for the simplest algorithms). To be safe they can inherit
elfi.Distribution
which defines the methods needed. In this case we
only need these for sampling, so implementing a static rvs
method
suffices. As was in the context of simulators, it is important to accept
the keyword argument random_state
, which is needed for ELFI’s
internal bookkeeping of pseudorandom number generation. Also the
size
keyword is needed (which in the simple cases is the same as the
batch_size
in the simulator definition).
# define prior for t1 as in Marin et al., 2012 with t1 in range [b, b]
class CustomPrior_t1(elfi.Distribution):
def rvs(b, size=1, random_state=None):
u = scipy.stats.uniform.rvs(loc=0, scale=1, size=size, random_state=random_state)
t1 = np.where(u<0.5, np.sqrt(2.*u)*bb, np.sqrt(2.*(1.u))*b+b)
return t1
# define prior for t2 conditionally on t1 as in Marin et al., 2012, in range [a, a]
class CustomPrior_t2(elfi.Distribution):
def rvs(t1, a, size=1, random_state=None):
locs = np.maximum(at1, t1a)
scales = a  locs
t2 = scipy.stats.uniform.rvs(loc=locs, scale=scales, size=size, random_state=random_state)
return t2
These indeed sample from a triangle:
t1_1000 = CustomPrior_t1.rvs(2, 1000)
t2_1000 = CustomPrior_t2.rvs(t1_1000, 1, 1000)
plt.scatter(t1_1000, t2_1000, s=4, edgecolor='none');
# plt.plot([0, 2, 2, 0], [1, 1, 1, 1], 'b') # outlines of the triangle
Let’s change the earlier priors to the new ones in the inference model:
t1.become(elfi.Prior(CustomPrior_t1, 2))
t2.become(elfi.Prior(CustomPrior_t2, t1, 1))
elfi.draw(d)
Note that t2
now depends on t1
. Yes, ELFI supports hierarchy.
Inference with rejection samplingÂ¶
The simplest ABC algorithm samples parameters from their prior distributions, runs the simulator with these and compares them to the observations. The samples are either accepted or rejected depending on how large the distance is. The accepted samples represent samples from the approximate posterior distribution.
In ELFI, ABC methods are initialized either with a node giving the
distance, or with the ElfiModel
object and the name of the distance
node. Depending on the inference method, additional arguments may be
accepted or required.
A common optional keyword argument, accepted by all inference methods,
batch_size
defines how many simulations are performed in each
passing through the graph.
Another optional keyword is the seed. This ensures that the outcome will be always the same for the same data and model. If you leave it out, a random seed will be taken.
rej = elfi.Rejection(d, batch_size=10000, seed=seed)
Note
In Python, doing many calculations with a single function call can potentially save a lot of CPU time, depending on the operation. For example, here we draw 10000 samples from t1, pass them as input to t2, draw 10000 samples from t2, and then use these both to run 10000 simulations and so forth. All this is done in one passing through the graph and hence the overall number of function calls is reduced 10000fold. However, this does not mean that batches should be as big as possible, since you may run out of memory, the fraction of time spent in function call overhead becomes insignificant, and many algorithms operate in multiples of batch_size. Furthermore, the batch_size is a crucial element for efficient parallelization (see the notebook on parallelization).
After the ABC method has been initialized, samples can be drawn from it.
By default, rejection sampling in ELFI works in quantile
mode i.e. a
certain quantile of the samples with smallest discrepancies is accepted.
The sample
method requires the number of output samples as a
parameter. Note that the simulator is then run (N/quantile)
times.
(Alternatively, the same behavior can be achieved by saying
n_sim=1000000
.)
The IPython magic command %time
is used here to give you an idea of
runtime on a typical personal computer. We will turn interactive
visualization on so that if you run this on a notebook you will see the
posterior forming from a prior distribution. In this case most of the
time is spent in drawing.
N = 1000
vis = dict(xlim=[2,2], ylim=[1,1])
# You can give the sample method a `vis` keyword to see an animation how the prior transforms towards the
# posterior with a decreasing threshold.
%time result = rej.sample(N, quantile=0.01, vis=vis)
CPU times: user 2.28 s, sys: 165 ms, total: 2.45 s
Wall time: 2.45 s
The sample
method returns a Sample
object, which contains
several attributes and methods. Most notably the attribute samples
contains an OrderedDict
(i.e. an ordered Python dictionary) of the
posterior numpy arrays for all the model parameters (elfi.Prior
s
in the model). For rejection sampling, other attributes include e.g. the
threshold
, which is the threshold value resulting in the requested
quantile.
result.samples['t1'].mean()
0.56
The Sample
object includes a convenient summary
method:
result.summary()
Method: Rejection
Number of samples: 1000
Number of simulations: 100000
Threshold: 0.117
Sample means: t1: 0.556, t2: 0.219
Rejection sampling can also be performed with using a threshold or total number of simulations. Let’s define here threshold. This means that all draws from the prior for which the generated distance is below the threshold will be accepted as samples. Note that the simulator will run as long as it takes to generate the requested number of samples.
%time result2 = rej.sample(N, threshold=0.2)
print(result2) # the Sample object's __str__ contains the output from summary()
CPU times: user 222 ms, sys: 40.3 ms, total: 263 ms
Wall time: 261 ms
Method: Rejection
Number of samples: 1000
Number of simulations: 40000
Threshold: 0.185
Sample means: t1: 0.555, t2: 0.223
Iterative advancingÂ¶
Often it may not be practical to wait to the end before investigating
the results. There may be time constraints or one may wish to check the
results at certain intervals. For this, ELFI provides an iterative
approach to advance the inference. First one sets the objective of the
inference and then calls the iterate
method.
Below is an example how to run the inference until the objective has been reached or a maximum of one second of time has been used.
# Request for 1M simulations.
rej.set_objective(1000, n_sim=1000000)
# We only have 1 sec of time and we are unsure if we will be finished by that time.
# So lets simulate as many as we can.
time0 = time.time()
time1 = time0 + 1
while not rej.finished and time.time() < time1:
rej.iterate()
# One could investigate the rej.state or rej.extract_result() here
# to make more complicated stopping criterions
# Extract and print the result as it stands. It will show us how many simulations were generated.
print(rej.extract_result())
Method: Rejection
Number of samples: 1000
Number of simulations: 190000
Threshold: 0.0855
Sample means: t1: 0.561, t2: 0.218
# We will see that it was not finished in 1 sec
rej.finished
False
We could continue from this stage just by continuing to call the
iterate
method. The extract_result
will always give a proper
result even if the objective was not reached.
Next we will look into how to store all the data that was generated so far. This allows us to e.g. save the data to disk and continue the next day, or modify the model and reuse some of the earlier data if applicable.
Storing simulated dataÂ¶
As the samples are already in numpy arrays, you can just say e.g.
np.save('t1_data.npy', result.samples['t1'])
to save them. However,
ELFI provides some additional functionality. You may define a pool for
storing all outputs of any node in the model (not just the accepted
samples). Let’s save all outputs for t1
, t2
, S1
and S2
in our model:
pool = elfi.OutputPool(['t1', 't2', 'S1', 'S2'])
rej = elfi.Rejection(d, pool=pool)
%time result3 = rej.sample(N, n_sim=1000000)
result3
CPU times: user 5.26 s, sys: 37.1 ms, total: 5.3 s
Wall time: 5.3 s
Method: Rejection
Number of samples: 1000
Number of simulations: 1000000
Threshold: 0.036
Sample means: t1: 0.561, t2: 0.227
The benefit of the pool is that you may reuse simulations without having to resimulate them. Above we saved the summaries to the pool, so we can change the distance node of the model without having to resimulate anything. Let’s do that.
# Replace the current distance with a cityblock (manhattan) distance and recreate the inference
d.become(elfi.Distance('cityblock', S1, S2, p=1))
rej = elfi.Rejection(d, pool=pool)
%time result4 = rej.sample(N, n_sim=1000000)
result4
CPU times: user 636 ms, sys: 1.35 ms, total: 638 ms
Wall time: 638 ms
Method: Rejection
Number of samples: 1000
Number of simulations: 1000000
Threshold: 0.0452
Sample means: t1: 0.56, t2: 0.228
Note the significant saving in time, even though the total number of considered simulations stayed the same.
We can also continue the inference by increasing the total number of simulations and only have to simulate the new ones:
%time result5 = rej.sample(N, n_sim=1200000)
result5
CPU times: user 1.72 s, sys: 10.6 ms, total: 1.73 s
Wall time: 1.73 s
Method: Rejection
Number of samples: 1000
Number of simulations: 1200000
Threshold: 0.0417
Sample means: t1: 0.561, t2: 0.225
Above the results were saved into a python dictionary. If you store a lot of data to dictionaries, you will eventually run out of memory. ELFI provides an alternative pool that, by default, saves the outputs to standard numpy .npy files:
arraypool = elfi.ArrayPool(['t1', 't2', 'Y', 'd'])
rej = elfi.Rejection(d, pool=arraypool)
%time result5 = rej.sample(100, threshold=0.3)
CPU times: user 25.8 ms, sys: 3.27 ms, total: 29 ms
Wall time: 28.5 ms
This stores the simulated data in binary npy
format under
arraypool.path
, and can be loaded with np.load
.
# Let's flush the outputs to disk (alternatively you can save or close the pool) so that we can read the .npy files.
arraypool.flush()
import os
print('Files in', arraypool.path, 'are', os.listdir(arraypool.path))
Files in pools/arraypool_3521077242 are ['d.npy', 't1.npy', 't2.npy', 'Y.npy']
Now lets load all the parameters t1
that were generated with numpy:
np.load(arraypool.path + '/t1.npy')
array([ 0.79, 0.01, 1.47, ..., 0.98, 0.18, 0.5 ])
We can also close (or save) the whole pool if we wish to continue later:
arraypool.close()
name = arraypool.name
print(name)
arraypool_3521077242
And open it up later to continue where we were left. We can open it using its name:
arraypool = elfi.ArrayPool.open(name)
print('This pool has', len(arraypool), 'batches')
# This would give the contents of the first batch
# arraypool[0]
This pool has 3 batches
You can delete the files with:
arraypool.delete()
# verify the deletion
try:
os.listdir(arraypool.path)
except FileNotFoundError:
print("The directry is removed")
The directry is removed
Visualizing the resultsÂ¶
Instances of Sample
contain methods for some basic plotting (these
are convenience methods to plotting functions defined under
elfi.visualization
).
For example one can plot the marginal distributions:
result.plot_marginals();
Often “pairwise relationships” are more informative:
result.plot_pairs();
Note that if working in a noninteractive environment, you can use e.g.
plt.savefig('pairs.png')
after an ELFI plotting command to save the
current figure to disk.
Sequential Monte Carlo ABCÂ¶
Rejection sampling is quite inefficient, as it does not learn from its history. The sequential Monte Carlo (SMC) ABC algorithm does just that by applying importance sampling: samples are weighed according to the resulting discrepancies and the next population of samples is drawn near to the previous using the weights as probabilities.
For evaluating the weights, SMC ABC needs to be able to compute the
probability density of the generated parameters. In our MA2 example we
used custom priors, so we have to specify a pdf
function by
ourselves. If we used standard priors, this step would not be needed.
Let’s modify the prior distribution classes:
# define prior for t1 as in Marin et al., 2012 with t1 in range [b, b]
class CustomPrior_t1(elfi.Distribution):
def rvs(b, size=1, random_state=None):
u = scipy.stats.uniform.rvs(loc=0, scale=1, size=size, random_state=random_state)
t1 = np.where(u<0.5, np.sqrt(2.*u)*bb, np.sqrt(2.*(1.u))*b+b)
return t1
def pdf(x, b):
p = 1./b  np.abs(x) / (b*b)
p = np.where(p < 0., 0., p) # disallow values outside of [b, b] (affects weights only)
return p
# define prior for t2 conditionally on t1 as in Marin et al., 2012, in range [a, a]
class CustomPrior_t2(elfi.Distribution):
def rvs(t1, a, size=1, random_state=None):
locs = np.maximum(at1, t1a)
scales = a  locs
t2 = scipy.stats.uniform.rvs(loc=locs, scale=scales, size=size, random_state=random_state)
return t2
def pdf(x, t1, a):
locs = np.maximum(at1, t1a)
scales = a  locs
p = scipy.stats.uniform.pdf(x, loc=locs, scale=scales)
p = np.where(scales>0., p, 0.) # disallow values outside of [a, a] (affects weights only)
return p
# Redefine the priors
t1.become(elfi.Prior(CustomPrior_t1, 2, model=t1.model))
t2.become(elfi.Prior(CustomPrior_t2, t1, 1))
Run SMC ABCÂ¶
In ELFI, one can setup a SMC ABC sampler just like the Rejection sampler:
smc = elfi.SMC(d, batch_size=10000, seed=seed)
For sampling, one has to define the number of output samples, the number of populations and a schedule i.e. a list of quantiles to use for each population. In essence, a population is just refined rejection sampling.
N = 1000
schedule = [0.7, 0.2, 0.05]
%time result_smc = smc.sample(N, schedule)
INFO:elfi.methods.parameter_inference: Starting round 0 
INFO:elfi.methods.parameter_inference: Starting round 1 
INFO:elfi.methods.parameter_inference: Starting round 2 
CPU times: user 1.72 s, sys: 154 ms, total: 1.87 s
Wall time: 1.56 s
We can have summaries and plots of the results just like above:
result_smc.summary(all=True)
Method: SMC
Number of samples: 1000
Number of simulations: 170000
Threshold: 0.0493
Sample means: t1: 0.554, t2: 0.229
Population 0:
Method: Rejection within SMCABC
Number of samples: 1000
Number of simulations: 10000
Threshold: 0.488
Sample means: t1: 0.547, t2: 0.232
Population 1:
Method: Rejection within SMCABC
Number of samples: 1000
Number of simulations: 20000
Threshold: 0.172
Sample means: t1: 0.562, t2: 0.22
Population 2:
Method: Rejection within SMCABC
Number of samples: 1000
Number of simulations: 140000
Threshold: 0.0493
Sample means: t1: 0.554, t2: 0.229
Or just the means:
result_smc.sample_means_summary(all=True)
Sample means for population 0: t1: 0.547, t2: 0.232
Sample means for population 1: t1: 0.562, t2: 0.22
Sample means for population 2: t1: 0.554, t2: 0.229
result_smc.plot_marginals(all=True, bins=25, figsize=(8, 2), fontsize=12)
Obviously one still has direct access to the samples as well, which allows custom plotting:
n_populations = len(schedule)
fig, ax = plt.subplots(ncols=n_populations, sharex=True, sharey=True, figsize=(16,6))
for i, pop in enumerate(result_smc.populations):
s = pop.samples
ax[i].scatter(s['t1'], s['t2'], s=5, edgecolor='none');
ax[i].set_title("Population {}".format(i));
ax[i].plot([0, 2, 2, 0], [1, 1, 1, 1], 'b')
ax[i].set_xlabel('t1');
ax[0].set_ylabel('t2');
ax[0].set_xlim([2, 2])
ax[0].set_ylim([1, 1]);
It can be seen that the populations iteratively concentrate more and more around the true parameter values. Note, however, that samples from SMC are weighed, and the weights should be accounted for when interpreting the results. ELFI does this automatically when computing the mean, for example.
That’s it! See the other documentation for more advanced topics on e.g. BOLFI, external simulators and parallelization.
This tutorial is generated from a Jupyter notebook that can be found here.
ParallelizationÂ¶
Behind the scenes, ELFI can automatically parallelize the computational inference via different clients. Currently ELFI includes three clients:
elfi.clients.native
(activated by default): does not parallelize but makes it easy to test and debug your code.elfi.clients.multiprocessing
: basic local parallelization using Python’s builtin multiprocessing libraryelfi.clients.ipyparallel
: ipyparallel based client that can parallelize from multiple cores up to a distributed cluster.
A client is activated by giving the name of the client to
elfi.set_client
.
This tutorial shows how to activate and use the multiprocessing
or
ipyparallel
client with ELFI. The ipyparallel
client supports
parallelization from local computer up to a cluster environment. For
local parallelization however, the multiprocessing
client is simpler
to use. Let’s begin by importing ELFI and our example MA2 model from the
tutorial.
import elfi
from elfi.examples import ma2
Let’s get the model and plot it (requires graphviz)
model = ma2.get_model()
elfi.draw(model)
Multiprocessing clientÂ¶
The multiprocessing client allows you to easily use the cores available in your computer. You can activate it simply by
elfi.set_client('multiprocessing')
Any inference instance created after you have set the new client will automatically use it to perform the computations. Let’s try it with our MA2 example model from the tutorial. When running the next command, take a look at the system monitor of your operating system; it should show that all of your cores are doing heavy computation simultaneously.
rej = elfi.Rejection(model, 'd', batch_size=10000, seed=20170530)
%time result = rej.sample(5000, n_sim=int(1e6)) # 1 million simulations
CPU times: user 272 ms, sys: 28 ms, total: 300 ms
Wall time: 2.41 s
And that is it. The result object is also just like in the basic case:
# Print the summary
result.summary()
import matplotlib.pyplot as plt
result.plot_pairs();
plt.show()
Method: Rejection
Number of samples: 5000
Number of simulations: 1000000
Threshold: 0.0817
Sample means: t1: 0.68, t2: 0.133
Ipyparallel clientÂ¶
The ipyparallel
client allows you to parallelize the computations to
cluster environments. To use the ipyparallel
client, you first have
to create an ipyparallel
cluster. Below is an example of how to
start a local cluster to the background using 4 CPU cores:
!ipcluster start n 4 daemon
# This is here just to ensure that ipcluster has enough time to start properly before continuing
import time
time.sleep(10)
Note
The exclamation mark above is a Jupyter syntax for executing shell commands. You can run the same command in your terminal without the exclamation mark.
Tip
Please see the ipyparallel documentation (https://ipyparallel.readthedocs.io/en/latest/intro.html#gettingstarted) for more information and details for setting up and using ipyparallel clusters in different environments.
Running parallel inference with ipyparallelÂ¶
After the cluster has been set up, we can proceed as usual. ELFI will take care of the parallelization from now on:
# Let's start using the ipyparallel client
elfi.set_client('ipyparallel')
rej = elfi.Rejection(model, 'd', batch_size=10000, seed=20170530)
%time result = rej.sample(5000, n_sim=int(5e6)) # 5 million simulations
CPU times: user 3.16 s, sys: 184 ms, total: 3.35 s
Wall time: 13.4 s
To summarize, the only thing that needed to be changed from the basic
scenario was creating the ipyparallel
cluster and enabling the
ipyparallel
client.
Working interactively with ipyparallelÂ¶
If you are using the ipyparallel
client from an interactive
environment (e.g. jupyter notebook) there are some things to take care
of. All imports and definitions must be visible to all ipyparallel
engines. You can ensure this by writing a script file that has all the
definitions in it. In a distributed setting, this file must be present
in all remote workers running an ipyparallel
engine.
However, you may wish to experiment in an interactive session, using
e.g. a jupyter notebook. ipyparallel
makes it possible to
interactively define functions for ELFI model and send them to workers.
This is especially useful if you work from a jupyter notebook. We will
show a few examples. More information can be found from `ipyparallel
documentation <http://ipyparallel.readthedocs.io/>`__.
In interactive sessions, you can change the model with builtin functionality without problems:
d2 = elfi.Distance('cityblock', model['S1'], model['S2'], p=1)
rej2 = elfi.Rejection(d2, batch_size=10000)
result2 = rej2.sample(1000, quantile=0.01)
But let’s say you want to use your very own distance function in a jupyter notebook:
def my_distance(x, y):
# Note that interactively defined functions must use full module names, e.g. numpy instead of np
return numpy.sum((xy)**2, axis=1)
d3 = elfi.Distance(my_distance, model['S1'], model['S2'])
rej3 = elfi.Rejection(d3, batch_size=10000)
This function definition is not automatically visible for the
ipyparallel
engines if it is not defined in a physical file. The
engines run in different processes and will not see interactively
defined objects and functions. The below would therefore fail:
# This will fail if you try it!
# result3 = rej3.sample(1000, quantile=0.01)
Ipyparallel provides a way to manually push
the new definition to
the scopes of the engines from interactive sessions. Because
my_distance
also uses numpy
, that must be imported in the
engines as well:
# Get the ipyparallel client
ipyclient = elfi.get_client().ipp_client
# Import numpy in the engines (note that you cannot use "as" abbreviations, but must use plain imports)
with ipyclient[:].sync_imports():
import numpy
# Then push my_distance to the engines
ipyclient[:].push({'my_distance': my_distance});
importing numpy on engine(s)
The above may look a bit cumbersome, but now this works:
rej3.sample(1000, quantile=0.01) # now this works
Method: Rejection
Number of samples: 1000
Number of simulations: 100000
Threshold: 0.0136
Sample means: t1: 0.676, t2: 0.129
However, a simpler solution to cases like this may be to define your
functions in external scripts (see elfi.examples.ma2
) and have the
module files be available in the folder where you run your ipyparallel
engines.
Remember to stop the ipcluster when doneÂ¶
!ipcluster stop
20170719 16:20:58.662 [IPClusterStop] Stopping cluster [pid=21020] with [signal=<Signals.SIGINT: 2>]
This tutorial is generated from a Jupyter notebook that can be found here.
BOLFIÂ¶
In practice inference problems often have a complicated and computationally heavy simulator, and one simply cannot run it for millions of times. The Bayesian Optimization for LikelihoodFree Inference BOLFI framework is likely to prove useful in such situation: a statistical model (usually Gaussian process, GP) is created for the discrepancy, and its minimum is inferred with Bayesian optimization. This approach typically reduces the number of required simulator calls by several orders of magnitude.
This tutorial demonstrates how to use BOLFI to do LFI in ELFI.
import numpy as np
import scipy.stats
import matplotlib
import matplotlib.pyplot as plt
%matplotlib inline
%precision 2
import logging
logging.basicConfig(level=logging.INFO)
# Set an arbitrary global seed to keep the randomly generated quantities the same
seed = 1
np.random.seed(seed)
import elfi
Although BOLFI is best used with complicated simulators, for demonstration purposes we will use the familiar MA2 model introduced in the basic tutorial, and load it from readymade examples:
from elfi.examples import ma2
model = ma2.get_model(seed_obs=seed)
elfi.draw(model)
Fitting the surrogate modelÂ¶
Now we can immediately proceed with the inference. However, when dealing with a Gaussian process, it may be beneficial to take a logarithm of the discrepancies in order to reduce the effect that high discrepancies have on the GP. (Sometimes you may want to add a small constant to avoid very negative or even Inf distances occurring especially if it is likely that there can be exact matches between simulated and observed data.) In ELFI such transformed node can be created easily:
log_d = elfi.Operation(np.log, model['d'])
As BOLFI is a more advanced inference method, its interface is also a
bit more involved as compared to for example rejection sampling. But not
much: Using the same graphical model as earlier, the inference could
begin by defining a Gaussian process (GP) model, for which ELFI uses the
GPy library. This could be
given as an elfi.GPyRegression
object via the keyword argument
target_model
. In this case, we are happy with the default that ELFI
creates for us when we just give it each parameter some bounds
as a
dictionary.
Other notable arguments include the initial_evidence
, which gives
the number of initialization points sampled straight from the priors
before starting to optimize the acquisition of points,
update_interval
which defines how often the GP hyperparameters are
optimized, and acq_noise_var
which defines the diagonal covariance
of noise added to the acquired points. Note that in general BOLFI does
not benefit from a batch_size
higher than one, since the acquisition
surface is updated after each batch (especially so if the noise is 0!).
bolfi = elfi.BOLFI(log_d, batch_size=1, initial_evidence=20, update_interval=10,
bounds={'t1':(2, 2), 't2':(1, 1)}, acq_noise_var=[0.1, 0.1], seed=seed)
Sometimes you may have some samples readily available. You could then
initialize the GP model with a dictionary of previous results by giving
initial_evidence=result.outputs
.
The BOLFI class can now try to fit
the surrogate model (the GP) to
the relationship between parameter values and the resulting
discrepancies. We’ll request only 100 evidence points (including the
initial_evidence
defined above).
%time post = bolfi.fit(n_evidence=200)
INFO:elfi.methods.parameter_inference:BOLFI: Fitting the surrogate model...
INFO:elfi.methods.posteriors:Using optimized minimum value (1.6146) of the GP discrepancy mean function as a threshold
CPU times: user 1min 48s, sys: 1.29 s, total: 1min 50s
Wall time: 1min
(More on the returned BolfiPosterior
object
below.)
Note that in spite of the very few simulator runs, fitting the model took longer than any of the previous methods. Indeed, BOLFI is intended for scenarios where the simulator takes a lot of time to run.
The fitted target_model
uses the GPy library, and can be
investigated further:
bolfi.target_model
Name : GP regression
Objective : 151.86636065302943
Number of Parameters : 4
Number of Optimization Parameters : 4
Updates : True
Parameters:
[1mGP_regression. [0;0m  value  constraints  priors
[1msum.rbf.variance [0;0m  0.321697451372  +ve  Ga(0.024, 1)
[1msum.rbf.lengthscale [0;0m  0.541352150083  +ve  Ga(1.3, 1)
[1msum.bias.variance [0;0m  0.021827430988  +ve  Ga(0.006, 1)
[1mGaussian_noise.variance[0;0m  0.183562040169  +ve 
bolfi.plot_state();
<matplotlib.figure.Figure at 0x11b2b2ba8>
It may be useful to see the acquired parameter values and the resulting discrepancies:
bolfi.plot_discrepancy();
There could be an unnecessarily high number of points at parameter
bounds. These could probably be decreased by lowering the covariance of
the noise added to acquired points, defined by the optional
acq_noise_var
argument for the BOLFI constructor. Another
possibility could be to add virtual derivative observations at the
borders, though not yet
implemented in ELFI.
BOLFI PosteriorÂ¶
Above, the fit
method returned a BolfiPosterior
object
representing a BOLFI posterior (please see the
paper for
details). The fit
method accepts a threshold parameter; if none is
given, ELFI will use the minimum value of discrepancy estimate mean.
Afterwards, one may request for a posterior with a different threshold:
post2 = bolfi.extract_posterior(1.)
One can visualize a posterior directly (remember that the priors form a triangle):
post.plot(logpdf=True)
SamplingÂ¶
Finally, samples from the posterior can be acquired with an MCMC sampler. By default it runs 4 chains, and half of the requested samples are spent in adaptation/warmup. Note that depending on the smoothness of the GP approximation, the number of priors, their gradients etc., this may be slow.
%time result_BOLFI = bolfi.sample(1000, info_freq=1000)
INFO:elfi.methods.posteriors:Using optimized minimum value (1.6146) of the GP discrepancy mean function as a threshold
INFO:elfi.methods.mcmc:NUTS: Performing 1000 iterations with 500 adaptation steps.
INFO:elfi.methods.mcmc:NUTS: Adaptation/warmup finished. Sampling...
INFO:elfi.methods.mcmc:NUTS: Acceptance ratio: 0.423. After warmup 68 proposals were outside of the region allowed by priors and rejected, decreasing acceptance ratio.
INFO:elfi.methods.mcmc:NUTS: Performing 1000 iterations with 500 adaptation steps.
INFO:elfi.methods.mcmc:NUTS: Adaptation/warmup finished. Sampling...
INFO:elfi.methods.mcmc:NUTS: Acceptance ratio: 0.422. After warmup 71 proposals were outside of the region allowed by priors and rejected, decreasing acceptance ratio.
INFO:elfi.methods.mcmc:NUTS: Performing 1000 iterations with 500 adaptation steps.
INFO:elfi.methods.mcmc:NUTS: Adaptation/warmup finished. Sampling...
INFO:elfi.methods.mcmc:NUTS: Acceptance ratio: 0.419. After warmup 65 proposals were outside of the region allowed by priors and rejected, decreasing acceptance ratio.
INFO:elfi.methods.mcmc:NUTS: Performing 1000 iterations with 500 adaptation steps.
INFO:elfi.methods.mcmc:NUTS: Adaptation/warmup finished. Sampling...
INFO:elfi.methods.mcmc:NUTS: Acceptance ratio: 0.439. After warmup 66 proposals were outside of the region allowed by priors and rejected, decreasing acceptance ratio.
4 chains of 1000 iterations acquired. Effective sample size and Rhat for each parameter:
t1 2222.1197791 1.00106816947
t2 2256.93599184 1.0003364409
CPU times: user 1min 45s, sys: 1.29 s, total: 1min 47s
Wall time: 55.1 s
The sampling algorithms may be finetuned with some parameters. The
default
NoUTurnSampler
is a sophisticated algorithm, and in some cases one may get warnings
about diverged proposals, which are signs that something may be wrong
and should be
investigated.
It is good to understand the cause of these warnings although they don’t
automatically mean that the results are unreliable. You could try
rerunning the sample
method with a higher target probability
target_prob
during adaptation, as its default 0.6 may be inadequate
for a nonsmooth posteriors, but this will slow down the sampling.
Note also that since MCMC proposals outside the region allowed by either the model priors or GP bounds are rejected, a tight domain may lead to suboptimal overall acceptance ratio. In our MA2 case the prior defines a triangleshaped uniform support for the posterior, making it a good example of a difficult model for the NUTS algorithm.
Now we finally have a Sample
object again, which has several
convenience methods:
result_BOLFI
Method: BOLFI
Number of samples: 2000
Number of simulations: 200
Threshold: 1.61
Sample means: t1: 0.429, t2: 0.0277
result_BOLFI.plot_traces();
The black vertical lines indicate the end of warmup, which by default is half of the number of iterations.
result_BOLFI.plot_marginals();
This tutorial is generated from a Jupyter notebook that can be found here.
Using nonPython operationsÂ¶
If your simulator or other operations are implemented in a programming language other than Python, you can still use ELFI. This notebook briefly demonstrates how to do this in three common scenarios:
 External executable (written e.g. in C++ or a shell script)
 R function
 MATLAB function
Let’s begin by importing some libraries that we will be using:
import os
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import scipy.io as sio
import scipy.stats as ss
import elfi
import elfi.examples.bdm
import elfi.examples.ma2
%matplotlib inline
Note
To run some parts of this notebook you need to either compile the simulator, have R or MATLAB installed and install their respective wrapper libraries.
External executablesÂ¶
ELFI supports using external simulators and other operations that can be called from the commandline. ELFI provides some tools to easily incorporate such operations to ELFI models. This functionality is introduced in this tutorial.
We demonstrate here how to wrap executables as ELFI nodes. We will first
use elfi.tools.external_operation
tool to wrap executables as a
Python callables (function). Let’s first investigate how it works with a
simple shell echo
command:
# Make an external command. {0} {1} are positional arguments and {seed} a keyword argument `seed`.
command = 'echo {0} {1} {seed}'
echo_sim = elfi.tools.external_operation(command)
# Test that `echo_sim` can now be called as a regular python function
echo_sim(3, 1, seed=123)
array([ 3., 1., 123.])
The placeholders for arguments in the command string are just Python’s
`format strings
<https://docs.python.org/3/library/string.html#formatstrings>`__.
Currently echo_sim
only accepts scalar arguments. In order to work
in ELFI, echo_sim
needs to be vectorized so that we can pass to it a
vector of arguments. ELFI provides a handy tool for this as well:
# Vectorize it with elfi tools
echo_sim_vec = elfi.tools.vectorize(echo_sim)
# Make a simple model
m = elfi.ElfiModel(name='echo')
elfi.Prior('uniform', .005, 2, model=m, name='alpha')
elfi.Simulator(echo_sim_vec, m['alpha'], 0, name='echo')
# Test to generate 3 simulations from it
m['echo'].generate(3)
array([[ 1.93678222e+00, 0.00000000e+00, 7.43529055e+08],
[ 9.43846120e01, 0.00000000e+00, 7.43529055e+08],
[ 2.67626618e01, 0.00000000e+00, 7.43529055e+08]])
So above, the first column draws from our uniform prior for \(\alpha\), the second column has constant zeros, and the last one lists the seeds provided to the command by ELFI.
Complex external operations \(\) case BDMÂ¶
To provide a more realistic example of external operations, we will consider the BirthDeathMutation (BDM) model used in *Lintusaari at al 2016* [1].
BirthDeathMutation processÂ¶
We will consider here the BirthDeathMutation process simulator introduced in Tanaka et al 2006 [2] for the spread of Tuberculosis. The simulator outputs a count vector where each of its elements represents a “mutation” of the disease and the count describes how many are currently infected by that mutation. There are three rates and the population size:
 \(\alpha\)  (birth rate) the rate at which any infectious host transmits the disease.
 \(\delta\)  (death rate) the rate at which any existing infectious hosts either recovers or dies.
 \(\tau\)  (mutation rate) the rate at which any infectious host develops a new unseen mutation of the disease within themselves.
 \(N\)  (population size) the size of the simulated infectious population
It is assumed that the susceptible population is infinite, the hosts carry only one mutation of the disease and transmit that mutation onward. A more accurate description of the model can be found from the original paper or e.g. *Lintusaari at al 2016* [1].
This simulator cannot be implemented effectively with vectorized operations so we have implemented it with C++ that handles loops efficiently. We will now reproduce Figure 6(a) in *Lintusaari at al 2016* [2] with ELFI. Let’s start by defining some constants:
# Fixed model parameters
delta = 0
tau = 0.198
N = 20
# The zeros are to make the observed population vector have length N
y_obs = np.array([6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], dtype='int16')
Let’s build the beginning of a new model for the birth rate \(\alpha\) as the only unknown
m = elfi.ElfiModel(name='bdm')
elfi.Prior('uniform', .005, 2, model=m, name='alpha')
Prior(name='alpha', 'uniform')
# Get the BDM source directory
sources_path = elfi.examples.bdm.get_sources_path()
# Compile (unixlike systems)
!make C $sources_path
# Move the executable in to the working directory
!mv $sources_path/bdm .
g++ bdm.cpp std=c++0x O Wall o bdm
Note
The source code for the BDM simulator comes with ELFI. You can get the directory with elfi.examples.bdm.get_source_directory(). Under unixlike systems it can be compiled with just typing make to console in the source directory. For windows systems, you need to have some C++ compiler available to compile it.
# Test the executable (assuming we have the executable `bdm` in the working directory)
sim = elfi.tools.external_operation('./bdm {0} {1} {2} {3} seed {seed} mode 1')
sim(1, delta, tau, N, seed=123)
array([ 19., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0.])
The BDM simulator is actually already internally vectorized if you
provide it an input file with parameters on the rows. This is more
efficient than looping in Python (elfi.tools.vectorize
), because one
simulation takes very little time and we wish to generate tens of
thousands of simulations. We will also here redirect the output to a
file and then read the file into a numpy array.
This is just one possibility among the many to implement this. The most efficient would be to write a native Python module with C++ but it’s beyond the scope of this article. So let’s work through files which is a fairly common situation especially with existing software.
# Assuming we have the executable `bdm` in the working directory
command = './bdm {filename} seed {seed} mode 1 > {output_filename}'
# Function to prepare the inputs for the simulator. We will create filenames and write an input file.
def prepare_inputs(*inputs, **kwinputs):
alpha, delta, tau, N = inputs
meta = kwinputs['meta']
# Organize the parameters to an array. The broadcasting works nicely with constant arguments here.
param_array = np.row_stack(np.broadcast(alpha, delta, tau, N))
# Prepare a unique filename for parallel settings
filename = '{model_name}_{batch_index}_{submission_index}.txt'.format(**meta)
np.savetxt(filename, param_array, fmt='%.4f %.4f %.4f %d')
# Add the filenames to kwinputs
kwinputs['filename'] = filename
kwinputs['output_filename'] = filename[:4] + '_out.txt'
# Return new inputs that the command will receive
return inputs, kwinputs
# Function to process the result of the simulation
def process_result(completed_process, *inputs, **kwinputs):
output_filename = kwinputs['output_filename']
# Read the simulations from the file.
simulations = np.loadtxt(output_filename, dtype='int16')
# Clean up the files after reading the data in
os.remove(kwinputs['filename'])
os.remove(output_filename)
# This will be passed to ELFI as the result of the command
return simulations
# Create the python function (do not read stdout since we will work through files)
bdm = elfi.tools.external_operation(command,
prepare_inputs=prepare_inputs,
process_result=process_result,
stdout=False)
Now let’s replace the echo simulator with this. To create unique but
informative filenames, we ask ELFI to provide the operation some meta
information. That will be available under the meta
keyword (see the
prepare_inputs
function above):
# Create the simulator
bdm_node = elfi.Simulator(bdm, m['alpha'], delta, tau, N, observed=y_obs, name='sim')
# Ask ELFI to provide the meta dict
bdm_node.uses_meta = True
# Draw the model
elfi.draw(m)
# Test it
data = bdm_node.generate(3)
print(data)
[[13 1 4 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[19 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[14 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]]
Completing the BDM modelÂ¶
We are now ready to finish up the BDM model. To reproduce Figure 6(a) in *Lintusaari at al 2016* [2], let’s add different summaries and discrepancies to the model and run the inference for each of them:
def T1(clusters):
clusters = np.atleast_2d(clusters)
return np.sum(clusters > 0, 1)/np.sum(clusters, 1)
def T2(clusters, n=20):
clusters = np.atleast_2d(clusters)
return 1  np.sum((clusters/n)**2, axis=1)
# Add the different distances to the model
elfi.Summary(T1, bdm_node, name='T1')
elfi.Distance('minkowski', m['T1'], p=1, name='d_T1')
elfi.Summary(T2, bdm_node, name='T2')
elfi.Distance('minkowski', m['T2'], p=1, name='d_T2')
elfi.Distance('minkowski', m['sim'], p=1, name='d_sim')
Distance(name='d_sim')
elfi.draw(m)
# Save parameter and simulation results in memory to speed up the later inference
pool = elfi.OutputPool(['alpha', 'sim'])
# Fix a seed
seed = 20170511
rej = elfi.Rejection(m, 'd_T1', batch_size=10000, pool=pool, seed=seed)
%time T1_res = rej.sample(5000, n_sim=int(1e5))
rej = elfi.Rejection(m, 'd_T2', batch_size=10000, pool=pool, seed=seed)
%time T2_res = rej.sample(5000, n_sim=int(1e5))
rej = elfi.Rejection(m, 'd_sim', batch_size=10000, pool=pool, seed=seed)
%time sim_res = rej.sample(5000, n_sim=int(1e5))
CPU times: user 3.11 s, sys: 143 ms, total: 3.26 s
Wall time: 5.56 s
CPU times: user 29.9 ms, sys: 1.45 ms, total: 31.3 ms
Wall time: 31.2 ms
CPU times: user 33.8 ms, sys: 500 Âµs, total: 34.3 ms
Wall time: 34 ms
# Load a precomputed posterior based on an analytic solution (see Lintusaari et al 2016)
matdata = sio.loadmat('./resources/bdm.mat')
x = matdata['likgrid'].reshape(1)
posterior_at_x = matdata['post'].reshape(1)
# Plot the reference
plt.figure()
plt.plot(x, posterior_at_x, c='k')
# Plot the different curves
for res, d_node, c in ([sim_res, 'd_sim', 'b'], [T1_res, 'd_T1', 'g'], [T2_res, 'd_T2', 'r']):
alphas = res.outputs['alpha']
dists = res.outputs[d_node]
# Use gaussian kde to make the curves look nice. Note that this tends to benefit the algorithm 1
# a lot as it ususally has only a very few accepted samples with 100000 simulations
kde = ss.gaussian_kde(alphas[dists<=0])
plt.plot(x, kde(x), c=c)
plt.legend(['reference', 'algorithm 1', 'algorithm 2, T1\n(eps=0)', 'algorithm 2, T2\n(eps=0)'])
plt.xlim([.2, 1.2]);
print('Results after 100000 simulations. Compare to figure 6(a) in Lintusaari et al. 2016.')
Results after 100000 simulations. Compare to figure 6(a) in Lintusaari et al. 2016.
Interfacing with RÂ¶
It is possible to run R scripts in command line for example with
Rscript.
However, in Python it may be more convenient to use
rpy2, which allows convenient access to
the functionality of R from within Python. You can install it with
pip install rpy2
.
Here we demonstrate how to calculate the summary statistics used in the
ELFI tutorial (autocovariances) using R’s acf
function for the MA2
model.
import rpy2.robjects as robj
from rpy2.robjects import numpy2ri as np2ri
# Converts numpy arrays automatically
np2ri.activate()
Note
See this issue if you get a undefined symbol: PC error in the import after installing rpy2 and you are using Anaconda.
Let’s create a Python function that wraps the R commands (please see the documentation of rpy2 for details):
robj.r('''
# create a function `f`
f < function(x, lag=1) {
ac = acf(x, plot=FALSE, type="covariance", lag.max=lag, demean=FALSE)
ac[['acf']][lag+1]
}
''')
f = robj.globalenv['f']
def autocovR(x, lag=1):
x = np.atleast_2d(x)
apply = robj.r['apply']
ans = apply(x, 1, f, lag=lag)
return np.atleast_1d(ans)
# Test it
autocovR(np.array([[1,2,3,4], [4,5,6,7]]), 1)
array([ 5., 23.])
Load a ready made MA2 model:
ma2 = elfi.examples.ma2.get_model(seed_obs=4)
elfi.draw(ma2)
Replace the summaries S1 and S2 with our R autocovariance function.
# Replace with R autocov
S1 = elfi.Summary(autocovR, ma2['MA2'], 1)
S2 = elfi.Summary(autocovR, ma2['MA2'], 2)
ma2['S1'].become(S1)
ma2['S2'].become(S2)
# Run the inference
rej = elfi.Rejection(ma2, 'd', batch_size=1000, seed=seed)
rej.sample(100)
Method: Rejection
Number of samples: 100
Number of simulations: 10000
Threshold: 0.111
Sample means: t1: 0.599, t2: 0.177
Interfacing with MATLABÂ¶
There are a number of options for running MATLAB (or Octave) scripts from within Python. Here, evaluating the distance is demonstrated with a MATLAB function using the official MATLAB Python cd API. (Tested with MATLAB 2016b.)
import matlab.engine
A MATLAB session needs to be started (and stopped) separately:
eng = matlab.engine.start_matlab() # takes a while...
Similarly as with R, we have to write a piece of code to interface between MATLAB and Python:
def euclidean_M(x, y):
# MATLAB array initialized with Python's list
ddM = matlab.double((xy).tolist())
# euclidean distance
dM = eng.sqrt(eng.sum(eng.power(ddM, 2.0), 2))
# Convert back to numpy array
d = np.atleast_1d(dM).reshape(1)
return d
# Test it
euclidean_M(np.array([[1,2,3], [6,7,8], [2,2,3]]), np.array([2,2,2]))
array([ 1.41421356, 8.77496439, 1. ])
Load a ready made MA2 model:
ma2M = elfi.examples.ma2.get_model(seed_obs=4)
elfi.draw(ma2M)
Replace the summaries S1 and S2 with our R autocovariance function.
# Replace with Matlab distance implementation
d = elfi.Distance(euclidean_M, ma2M['S1'], ma2M['S2'])
ma2M['d'].become(d)
# Run the inference
rej = elfi.Rejection(ma2M, 'd', batch_size=1000, seed=seed)
rej.sample(100)
Method: Rejection
Number of samples: 100
Number of simulations: 10000
Threshold: 0.113
Sample means: t1: 0.602, t2: 0.178
Finally, don’t forget to quit the MATLAB session:
eng.quit()
VerdictÂ¶
We showed here a few examples of how to incorporate non Python operations to ELFI models. There are multiple other ways to achieve the same results and even make the wrapping more efficient.
Wrapping often introduces some overhead to the evaluation of the generative model. In many cases however this is not an issue since the operations are usually expensive by themselves making the added overhead insignificant.
ReferencesÂ¶
 [1] Jarno Lintusaari, Michael U. Gutmann, Ritabrata Dutta, Samuel Kaski, Jukka Corander; Fundamentals and Recent Developments in Approximate Bayesian Computation. Syst Biol 2017; 66 (1): e66e82. doi: 10.1093/sysbio/syw077
 [2] Tanaka, Mark M., et al. “Using approximate Bayesian computation to estimate tuberculosis transmission parameters from genotype data.” Genetics 173.3 (2006): 15111520.
Implementing a new inference methodÂ¶
This tutorial provides the fundamentals for implementing custom parameter inference methods using ELFI. ELFI provides many features out of the box, such as parallelization or random state handling. In a typical case these happen “automatically” behind the scenes when the algorithms are built on top of the provided interface classes.
The base class for parameter inference classes is the ParameterInference interface
which is found from the elfi.methods.parameter_inference
module. Among the methods in
the interface, those that must be implemented raise a NotImplementedError
. In
addition, you probably also want to override at least the update
and __init__
methods.
Let’s create an empty skeleton for a custom method that includes just the minimal set of methods to create a working algorithm in ELFI:
from elfi.methods.parameter_inference import ParameterInference
class CustomMethod(ParameterInference):
def __init__(self, model, output_names, **kwargs):
super(CustomMethod, self).__init__(model, output_names, **kwargs)
def set_objective(self):
# Request 3 batches to be generated
self.objective['n_batches'] = 3
def extract_result(self):
return self.state
The method extract_result
is called by ELFI in the end of inference and should return
a ParameterInferenceResult
object (elfi.methods.result
module). For illustration
we will however begin by returning the member state
dictionary. It stores all the
current state information of the inference. Let’s make an instance of our method and run
it:
import elfi.examples.ma2 as ma2
# Get a ready made MA2 model to test our inference method with
m = ma2.get_model()
# We want the outputs from node 'd' of the model `m` to be available
custom_method = CustomMethod(m, ['d'])
# Run the inference
custom_method.infer() # {'n_batches': 3, 'n_sim': 3000}
Running the above returns the state dictionary. We will find a few keys in it that track
some basic properties of the state, such as the n_batches
telling how many batches has
been generated and n_sim
that tells the number of total simulations contained in those
batches. It should be n_batches
times the current batch size
(custom_method.batch_size
which was 1000 here by default).
You will find that the n_batches
in the state dictionary had a value 3. This is
because in our CustomMethod.set_objective
method, we set the n_batches
key of the
objective dictionary to that value. Every ParameterInference instance has a Python
dictionary called objective
that is a counterpart to the state
dictionary. The
objective defines the conditions when the inference is finished. The default controlling
key in that dictionary is the string n_batches
whose value tells ELFI how many batches
we need to generate in total from the provided generative ElfiModel
model. Inference
is considered finished when the n_batches
in the state
matches or exceeds that in
the objective
. The generation of batches is automatically parallelized in the
background, so we don’t have to worry about it.
Note
A batch in ELFI is a dictionary that maps names of nodes of the generative model to their outputs. An output in the batch consists of one or more runs of it’s operation stored to a numpy array. Each batch has an index, and the outputs in the same batch are guaranteed to be the same if you recompute the batch.
The algorithm, however, does nothing else at this point besides generating the 3 batches.
To actually do something with the batches, we can add the update
method that allows us
to update the state dictionary of the inference with any custom values. It takes in the
generated batch
dictionary and it’s index and is called by ELFI every time a new batch
is received. Let’s say we wish to filter parameters by a threshold (as in ABC Rejection
sampling) from the total number of simulations:
class CustomMethod(ParameterInference):
def __init__(self, model, output_names, **kwargs):
super(CustomMethod, self).__init__(model, output_names, **kwargs)
# Hard code a threshold and discrepancy node name for now
self.threshold = .1
self.discrepancy_name = output_names[0]
# Prepare lists to push the filtered outputs into
self.state['filtered_outputs'] = {name: [] for name in output_names}
def update(self, batch, batch_index):
super(CustomMethod, self).update(batch, batch_index)
# Make a filter mask (logical numpy array) from the distance array
filter_mask = batch[self.discrepancy_name] <= self.threshold
# Append the filtered parameters to their lists
for name in self.output_names:
values = batch[name]
self.state['filtered_outputs'][name].append(values[filter_mask])
... # other methods as before
m = ma2.get_model()
custom_method = CustomMethod(m, ['d'])
custom_method.infer() # {'n_batches': 3, 'n_sim': 3000, 'filtered_outputs': ...}
After running this you should have in the returned state dictionary the
filtered_outputs
key containing filtered distances for node d
from the 3 batches.
Note
The reason for the imposed structure in ParameterInference
is to encourage a
design where one can advance the inference iteratively using the iterate
method.
This makes it possible to stop at any point, check the current state and to be able to
continue. This is important as there are usually many moving parts, such as summary
statistic choices or deciding a good discrepancy function.
Now to be useful, we should allow the user to set the different options  the 3 batches is
not going to take her very far. The user also probably thinks in terms of simulations
rather than batches. ELFI allows you to replace the n_batches
with n_sim
key
in the objective to spare you from turning n_sim
to n_batches
in the code. Just
note that the n_sim
in the state will always be in multiples of the batch_size
.
Let’s modify the algorithm so, that the user can pass the threshold, the name of the discrepancy node and the number of simulations. And let’s also add the parameters to the outputs:
class CustomMethod(ParameterInference):
def __init__(self, model, discrepancy_name, threshold, **kwargs):
# Create a name list of nodes whose outputs we wish to receive
output_names = [discrepancy_name] + model.parameter_names
super(CustomMethod, self).__init__(model, output_names, **kwargs)
self.threshold = threshold
self.discrepancy_name = discrepancy_name
# Prepare lists to push the filtered outputs into
self.state['filtered_outputs'] = {name: [] for name in output_names}
def set_objective(self, n_sim):
self.objective['n_sim'] = n_sim
... # other methods as before
# Run it
custom_method = CustomMethod(m, 'd', threshold=.1, batch_size=1000)
custom_method.infer(n_sim=2000) # {'n_batches': 2, 'n_sim': 2000, 'filtered_outputs': ...}
Calling the inference method now returns the state dictionary that has also the filtered
parameters in it from each of the batches. Note that any arguments given to the infer
method are passed to the set_objective
method.
Now due to the structure of the algorithm the user can immediately continue from this state:
# Continue inference from the previous state (with n_sim=2000)
custom_method.infer(n_sim=4000) # {'n_batches': 4, 'n_sim': 4000, 'filtered_outputs': ...}
# Or use it iteratively
custom_method.set_objective(n_sim=6000)
custom_method.iterate()
assert custom_method.finished == False
# Investigate the current state
custom_method.extract_result() # {'n_batches': 5, 'n_sim': 5000, 'filtered_outputs': ...}
self.iterate()
assert custom_method.finished
custom_method.extract_result() # {'n_batches': 6, 'n_sim': 6000, 'filtered_outputs': ...}
This works, because the state is stored into the custom_method
instance, and we only
change the objective. Also ELFI calls iterate
internally in the infer
method.
The last finishing touch to our algorithm is to convert the state
dict to a more user
friendly format in the extract_result
method. First we want to convert the list of
filtered arrays from the batches to a numpy array. We will then wrap the result to a
elfi.methods.results.Sample
object and return it instead of the state
dict. Below
is the final complete implementation of our inference method class:
import numpy as np
from elfi.methods.parameter_inference import ParameterInference
from elfi.methods.results import Sample
class CustomMethod(ParameterInference):
def __init__(self, model, discrepancy_name, threshold, **kwargs):
# Create a name list of nodes whose outputs we wish to receive
output_names = [discrepancy_name] + model.parameter_names
super(CustomMethod, self).__init__(model, output_names, **kwargs)
self.threshold = threshold
self.discrepancy_name = discrepancy_name
# Prepare lists to push the filtered outputs into
self.state['filtered_outputs'] = {name: [] for name in output_names}
def set_objective(self, n_sim):
self.objective['n_sim'] = n_sim
def update(self, batch, batch_index):
super(CustomMethod, self).update(batch, batch_index)
# Make a filter mask (logical numpy array) from the distance array
filter_mask = batch[self.discrepancy_name] <= self.threshold
# Append the filtered parameters to their lists
for name in self.output_names:
values = batch[name]
self.state['filtered_outputs'][name].append(values[filter_mask])
def extract_result(self):
filtered_outputs = self.state['filtered_outputs']
outputs = {name: np.concatenate(filtered_outputs[name]) for name in self.output_names}
return Sample(
method_name='CustomMethod',
outputs=outputs,
parameter_names=self.parameter_names,
discrepancy_name=self.discrepancy_name,
n_sim=self.state['n_sim'],
threshold=self.threshold
)
Running the inference with the above implementation should now produce an user friendly output:
Method: CustomMethod
Number of posterior samples: 82
Number of simulations: 10000
Threshold: 0.1
Posterior means: t1: 0.687, t2: 0.152
Where to go from hereÂ¶
When implementing your own method it is advisable to read the documentation of the
ParameterInference class. In addition we recommend reading the Rejection
, SMC
and/or BayesianOptimization
class implementations from the source for some more
advanced techniques. These methods feature e.g. how to inject values from outside into the
ELFI model (acquisition functions in BayesianOptimization), how to modify the user
provided model to get e.g. the pdf:s of the parameters (SMC) and so forth.
Good to knowÂ¶
ELFI guarantees that computing a batch with the same index will always produce the same
output given the same model and ComputationContext
object. The ComputationContext
object holds the batch size, seed for the PRNG, the pool object of precomputed batches
of nodes. If your method uses random quantities in the algorithm, please make sure
to use the seed attribute of ParameterInference
so that your results will be
consistent.
If you want to provide values for outputs of certain nodes from outside the generative
model, you can return them from prepare_new_batch
method. They will replace any
default value or operation in that node. This is used e.g. in BOLFI
where values from
the acquisition function replace values coming from the prior in the Bayesian optimization
phase.
The ParameterInference instance has also the following helper classes:
BatchHandler
Â¶
ParameterInference class instantiates a elfi.client.BatchHandler
helper class that
is set as the self.batches
member variable. This object is in essence a wrapper to the
Client
interface making it easier to work with batches that are in computation. Some
of the duties of BatchHandler
is to keep track of the current batch_index and of the
status of the batches that have been submitted. You often don’t need to interact with it
directly.
OutputPool
Â¶
elfi.store.OutputPool
serves a dual purpose:
1. It stores all the computed outputs of selected nodes
2. It provides those outputs when a batch is recomputed saving the need to recompute them.
Note however that reusing the values is not always possible. In sequential algorithms that decide their next parameter values based on earlier results, modifications to the ELFI model will invalidate the earlier data. On the other hand, Rejection sampling for instance allows changing any of the summaries or distances and still reuse e.g. the simulations. This is because all the parameter values will still come from the same priors.
Parameter inference base classÂ¶

class
elfi.methods.parameter_inference.
ParameterInference
(model, output_names, batch_size=1000, seed=None, pool=None, max_parallel_batches=None)[source]Â¶ A base class for parameter inference methods.

model
Â¶ elfi.ElfiModel – The ELFI graph used by the algorithm

output_names
Â¶ list – Names of the nodes whose outputs are included in the batches

client
Â¶ elfi.client.ClientBase – The batches are computed in the client

max_parallel_batches
Â¶ int

state
Â¶ dict – Stores any changing data related to achieving the objective. Must include a key
n_batches
for determining when the inference is finished.

objective
Â¶ dict – Holds the data for the algorithm to internally determine how many batches are still needed. You must have a key
n_batches
here. By default the algorithm finished when then_batches
in the state dictionary is equal or greater to the corresponding objective value.

batches
Â¶ elfi.client.BatchHandler – Helper class for submitting batches to the client and keeping track of their indexes.

pool
Â¶ elfi.store.OutputPool – Pool object for storing and reusing node outputs.
Construct the inference algorithm object.
If you are implementing your own algorithm do not forget to call super.
Parameters:  model (ElfiModel) – Model to perform the inference with.
 output_names (list) – Names of the nodes whose outputs will be requested from the ELFI graph.
 batch_size (int, optional) –
 seed (int, optional) – Seed for the data generation from the ElfiModel
 pool (OutputPool, optional) – OutputPool both stores and provides precomputed values for batches.
 max_parallel_batches (int, optional) – Maximum number of batches allowed to be in computation at the same time. Defaults to number of cores in the client

batch_size
Â¶ Return the current batch_size.

extract_result
()[source]Â¶ Prepare the result from the current state of the inference.
ELFI calls this method in the end of the inference to return the result.
Returns: result Return type: elfi.methods.result.Result

infer
(*args, vis=None, **kwargs)[source]Â¶ Set the objective and start the iterate loop until the inference is finished.
See the other arguments from the set_objective method.
Returns: result Return type: Sample

iterate
()[source]Â¶ Advance the inference by one iteration.
This is a way to manually progress the inference. One iteration consists of waiting and processing the result of the next batch in succession and possibly submitting new batches.
Notes
If the next batch is ready, it will be processed immediately and no new batches are submitted.
New batches are submitted only while waiting for the next one to complete. There will never be more batches submitted in parallel than the max_parallel_batches setting allows.
Returns: Return type: None

parameter_names
Â¶ Return the parameters to be inferred.

plot_state
(**kwargs)[source]Â¶ Plot the current state of the algorithm.
Parameters:  axes (matplotlib.axes.Axes (optional)) –
 figure (matplotlib.figure.Figure (optional)) –
 xlim – xaxis limits
 ylim – yaxis limits
 interactive (bool (default False)) – If true, uses IPython.display to update the cell figure
 close – Close figure in the end of plotting. Used in the end of interactive mode.
Returns: Return type: None

pool
Â¶ Return the output pool of the inference.

prepare_new_batch
(batch_index)[source]Â¶ Prepare values for a new batch.
ELFI calls this method before submitting a new batch with an increasing index batch_index. This is an optional method to override. Use this if you have a need do do preparations, e.g. in Bayesian optimization algorithm, the next acquisition points would be acquired here.
If you need provide values for certain nodes, you can do so by constructing a batch dictionary and returning it. See e.g. BayesianOptimization for an example.
Parameters: batch_index (int) – next batch_index to be submitted Returns: batch – Keys should match to node names in the model. These values will override any default values or operations in those nodes. Return type: dict or None

seed
Â¶ Return the seed of the inference.

set_objective
(*args, **kwargs)[source]Â¶ Set the objective of the inference.
This method sets the objective of the inference (values typically stored in the self.objective dict).
Returns: Return type: None

update
(batch, batch_index)[source]Â¶ Update the inference state with a new batch.
ELFI calls this method when a new batch has been computed and the state of the inference should be updated with it. It is also possible to bypass ELFI and call this directly to update the inference.
Parameters:  batch (dict) – dict with self.outputs as keys and the corresponding outputs for the batch as values
 batch_index (int) –
Returns: Return type: None

ELFI architectureÂ¶
Here we explain the internal representation of the ELFI model. This representation contains everything that is needed to generate data, but is separate from e.g. the inference methods or the data storages. This information is aimed for developers and is not essential for using ELFI. We assume the reader is quite familiar with Python and has perhaps already read some of ELFI’s source code.
The low level representation of the ELFI model is a networkx.DiGraph
with node names
as the nodes. The representation of the node is stored to the corresponding attribute
dictionary of the networkx.DiGraph
. We call this attribute dictionary the node state
dictionary. The networkx.DiGraph
representation can be found from
ElfiModel.source_net
. Before the ELFI model can be ran, it needs to be compiled and
loaded with data (e.g. observed data, precomputed data, batch index, batch size etc). The
compilation and loading of data is the responsibility of the Client
implementation and
makes it possible in essence to translate ElfiModel
to any kind of computational
backend. Finally the class Executor
is responsible for running the compiled and loaded
model and producing the outputs of the nodes.
A user typically creates this low level representation by working with subclasses of
NodeReference
. These are easy to use UI classes of ELFI such as the elfi.Simulator
or
elfi.Prior
. Under the hood they create proper node state dictionaries stored into the
source_net
. The callables such as simulators or summaries that the user provides to
these classes are called operations.
The model graph representationÂ¶
The source_net
is a directed acyclic graph (DAG) and holds the state dictionaries of the
nodes and the edges between the nodes. An edge represents a dependency. For example and
edge from a prior node to the simulator node represents that the simulator requires a
value from the prior to be able to run. The edge name corresponds to a parameter name for
the operation, with integer names interpreted as positional parameters.
In the standard compilation process, the source_net
is augmented with additional nodes
such as batch_size or random_state, that are then added as dependencies for those
operations that require them. In addition the state dicts will be turned into either a
runnable operation or a precomputed value.
The execution order of the nodes in the compiled graph follows the topological ordering of the DAG (dependency order) and is guaranteed to be the same every time. Note that because the default behaviour is that nodes share a random state, changing a node that uses a shared random state will affect the result of any later node in the ordering using the same random state, even if they would be independent based on the graph topology.
State dictionaryÂ¶
The state of a node is a Python dictionary. It describes the type of the node and any other relevant state information, such as the user provided callable operation (e.g. simulator or summary statistic) and any additional parameters the operation needs to be provided in the compilation.
The following are reserved keywords of the state dict that serve as instructions for the ELFI compiler. They begin with an underscore. Currently these are:
 _operation : callable
 Operation of the node producing the output. Can not be used if _output is present.
 _output : variable
 Constant output of the node. Can not be used if _operation is present.
 _class : class
 The subclass of
NodeReference
that created the state.  _stochastic : bool, optional
 Indicates that the node is stochastic. ELFI will provide a random_state argument for such nodes, which contains a RandomState object for drawing random quantities. This node will appear in the computation graph. Using ELFI provided random states makes it possible to have repeatable experiments in ELFI.
 _observable : bool, optional
 Indicates that there is observed data for this node or that it can be derived from the observed data. ELFI will create a corresponding observed node into the compiled graph. These nodes are dependencies of discrepancy nodes.
 _uses_batch_size : bool, optional
 Indicates that the node operation requires
batch_size
as input. A corresponding edge from batch_size node to this node will be added to the compiled graph.  _uses_meta : bool, optional
 Indicates that the node operation requires meta information dictionary about the
execution. This includes, model name, batch index and submission index.
Useful for e.g. creating informative and unique file names. If the operation is
vectorized with
elfi.tools.vectorize
, then alsoindex_in_batch
will be added to the meta information dictionary.  _uses_observed : bool, optional
 Indicates that the node requires the observed data of its parents in the source_net as input. ELFI will gather the observed values of its parents to a tuple and link them to the node as a named argument observed.
 _parameter : bool, optional
 Indicates that the node is a parameter node
The compilation and data loading phasesÂ¶
The compilation of the computation graph is separated from the loading of the data for
making it possible to reuse the compiled model. The subclasses of the Loader
class
take responsibility of injecting data to the nodes of the compiled model. Examples of
injected data are precomputed values from the OutputPool
, the current random_state
and
so forth.
ContributingÂ¶
Contributions are welcome, and they are greatly appreciated! Every little bit helps, and credit will always be given.
You can contribute in many ways:
Types of ContributionsÂ¶
Report BugsÂ¶
Report bugs at https://github.com/elfidev/elfi/issues.
If you are reporting a bug, please include:
 Your operating system name and version.
 Any details about your local setup that might be helpful in troubleshooting.
 Detailed steps to reproduce the bug.
Fix BugsÂ¶
Look through the GitHub issues for bugs. Anything tagged with “bug” and “help wanted” is open to whoever wants to implement it.
Implement FeaturesÂ¶
Look through the GitHub issues for features. Anything tagged with “enhancement” and “help wanted” is open to whoever wants to implement it.
Write DocumentationÂ¶
ELFI could always use more documentation, whether as part of the official ELFI docs, in docstrings, or even on the web in blog posts, articles, and such.
Submit FeedbackÂ¶
The best way to send feedback is to file an issue at https://github.com/elfidev/elfi/issues.
If you are proposing a feature:
 Explain in detail how it would work.
 Keep the scope as narrow as possible, to make it easier to implement.
 Remember that this is a volunteerdriven project, and that contributions are welcome :)
Get Started!Â¶
Ready to contribute? Here’s how to set up ELFI for local development.
Fork the elfi repo on GitHub.
Clone your fork locally:
$ git clone git@github.com:your_name_here/elfi.git
Install your local copy and the development requirements into a conda environment:
$ conda create n elfi python=3.5 numpy $ source activate elfi $ cd elfi $ make dev
Create a branch for local development:
$ git checkout b nameofyourbugfixorfeature
Now you can make your changes locally.
Follow the Style Guidelines
When you’re done making changes, check that your changes pass flake8 and the tests:
$ make lint $ make test
Also make sure that the docstrings of your code are formatted properly:
$ make docs
Commit your changes and push your branch to GitHub:
$ git add . $ git commit m "Your detailed description of your changes." $ git push origin nameofyourbugfixorfeature
Submit a pull request through the GitHub website.
Style GuidelinesÂ¶
The Python code in ELFI mostly follows PEP8, which is considered the defacto code style guide for Python. Lines should not exceed 100 characters.
Docstrings follow the NumPy style.
Pull Request GuidelinesÂ¶
Before you submit a pull request, check that it meets these guidelines:
 The pull request should include tests that will be run automatically using TravisCI.
 If the pull request adds functionality, the docs should be updated. Put your new functionality into a function with a docstring, and add the feature to the list in README.rst.
 The pull request should work for Python 3.5 and later. Check https://travisci.org/elfidev/elfi/pull_requests and make sure that the tests pass for all supported Python versions.
CitationÂ¶
If you wish to cite ELFI, please use the paper in arXiv:
@misc{1708.00707,
Author = {Jarno Lintusaari and Henri Vuollekoski and Antti KangasrÃ¤Ã¤siÃ¶ and Kusti SkytÃ©n and Marko JÃ¤rvenpÃ¤Ã¤ and Michael Gutmann and Aki Vehtari and Jukka Corander and Samuel Kaski},
Title = {ELFI: Engine for Likelihood Free Inference},
Year = {2017},
Eprint = {arXiv:1708.00707},
}