Inference¶

Sampling¶

Functions for MCMC sampling.

Generate posterior predictive samples from a model given a trace.

This is a vectorized alternative to the standard sample_posterior_predictive function. It aims to be as compatible as possible with the original API, and is significantly faster. Both posterior predictive sampling functions have some remaining issues, and we encourage users to verify agreement across the results of both functions for the time being.

Parameters

trace: MultiTrace, xarray.Dataset, InferenceData, or List of points (dictionary): Trace generated from MCMC sampling.
samples: int, optional: Number of posterior predictive samples to generate. Defaults to one posterior predictive sample per posterior sample, that is, the number of draws times the number of chains. It is not recommended to modify this value; when modified, some chains may not be represented in the posterior predictive sample.
model: Model (optional if in `with` context): Model used to generate trace
var_names: Iterable[str]: List of vars to sample.
keep_size: bool, optional: Force posterior predictive sample to have the same shape as posterior and sample stats data: (nchains, ndraws, ...).
random_seed: int: Seed for the random number generator.

Returns

samples: dict: Dictionary with the variable names as keys, and values numpy arrays containing posterior predictive samples.

pymc3.sampling.init_nuts(init='auto', chains=1, n_init=500000, model=None, random_seed=None, progressbar=True, jitter_max_retries=10, **kwargs)¶

Set up the mass matrix initialization for NUTS.

NUTS convergence and sampling speed is extremely dependent on the choice of mass/scaling matrix. This function implements different methods for choosing or adapting the mass matrix.

Parameters

initstr

Initialization method to use.

auto: Choose a default initialization method automatically. Currently, this is jitter+adapt_diag, but this can change in the future. If you depend on the exact behaviour, choose an initialization method explicitly.
adapt_diag: Start with a identity mass matrix and then adapt a diagonal based on the variance of the tuning samples. All chains use the test value (usually the prior mean) as starting point.
jitter+adapt_diag: Same as adapt_diag, but use test value plus a uniform jitter in [-1, 1] as starting point in each chain.
advi+adapt_diag: Run ADVI and then adapt the resulting diagonal mass matrix based on the sample variance of the tuning samples.
advi+adapt_diag_grad: Run ADVI and then adapt the resulting diagonal mass matrix based on the variance of the gradients during tuning. This is experimental and might be removed in a future release.
advi: Run ADVI to estimate posterior mean and diagonal mass matrix.
advi_map: Initialize ADVI with MAP and use MAP as starting point.
map: Use the MAP as starting point. This is discouraged.
adapt_full: Adapt a dense mass matrix using the sample covariances. All chains use the test value (usually the prior mean) as starting point.
jitter+adapt_full: Same as adapt_full, but use test value plus a uniform jitter in [-1, 1] as starting point in each chain.

chainsint

Number of jobs to start.

n_initint

Number of iterations of initializer. Only works for ‘ADVI’ init methods.

modelModel (optional if in with context)

progressbarbool

Whether or not to display a progressbar for advi sampling.

jitter_max_retriesint

Maximum number of repeated attempts (per chain) at creating an initial matrix with uniform jitter that yields a finite probability. This applies to jitter+adapt_diag and jitter+adapt_full init methods.

**kwargskeyword arguments

Extra keyword arguments are forwarded to pymc3.NUTS.

Returns

startpymc3.model.Point: Starting point for sampler
nuts_samplerpymc3.step_methods.NUTS: Instantiated and initialized NUTS sampler object

pymc3.sampling.iter_sample(draws: int, step, start: Optional[Dict[Any, Any]] = None, trace=None, chain=0, tune: Optional[int] = None, model: Optional[pymc3.model.Model] = None, random_seed: Optional[Union[int, List[int]]] = None, callback=None)¶

Generate a trace on each iteration using the given step method.

Multiple step methods ared supported via compound step methods. Returns the amount of time taken.

Parameters

drawsint: The number of samples to draw
stepfunction: Step function
startdict: Starting point in parameter space (or partial point). Defaults to trace.point(-1)) if there is a trace provided and model.test_point if not (defaults to empty dict)
tracebackend, list, or MultiTrace: This should be a backend instance, a list of variables to track, or a MultiTrace object with past values. If a MultiTrace object is given, it must contain samples for the chain number chain. If None or a list of variables, the NDArray backend is used.
chainint, optional: Chain number used to store sample in backend. If cores is greater than one, chain numbers will start here.
tuneint, optional: Number of iterations to tune, if applicable (defaults to None)
modelModel (optional if in with context)
random_seedint or list of ints, optional: A list is accepted if more if cores is greater than one.
callback :: A function which gets called for every sample from the trace of a chain. The function is called with the trace and the current draw and will contain all samples for a single trace. the draw.chain argument can be used to determine which of the active chains the sample is drawn from. Sampling can be interrupted by throwing a KeyboardInterrupt in the callback.

Yields

traceMultiTrace: Contains all samples up to the current iteration

Examples

for trace in iter_sample(500, step):
    ...

pymc3.sampling.sample(draws=1000, step=None, init='auto', n_init=200000, start=None, trace=None, chain_idx=0, chains=None, cores=None, tune=1000, progressbar=True, model=None, random_seed=None, discard_tuned_samples=True, compute_convergence_checks=True, callback=None, jitter_max_retries=10, *, return_inferencedata=None, idata_kwargs: Optional[dict] = None, mp_ctx=None, pickle_backend: str = 'pickle', **kwargs)¶

Draw samples from the posterior using the given step methods.

Multiple step methods are supported via compound step methods.

Parameters

drawsint

The number of samples to draw. Defaults to 1000. The number of tuned samples are discarded by default. See discard_tuned_samples.

initstr

Initialization method to use for auto-assigned NUTS samplers.

auto: Choose a default initialization method automatically. Currently, this is jitter+adapt_diag, but this can change in the future. If you depend on the exact behaviour, choose an initialization method explicitly.
adapt_diag: Start with a identity mass matrix and then adapt a diagonal based on the variance of the tuning samples. All chains use the test value (usually the prior mean) as starting point.
jitter+adapt_diag: Same as adapt_diag, but add uniform jitter in [-1, 1] to the starting point in each chain.
advi+adapt_diag: Run ADVI and then adapt the resulting diagonal mass matrix based on the sample variance of the tuning samples.
advi+adapt_diag_grad: Run ADVI and then adapt the resulting diagonal mass matrix based on the variance of the gradients during tuning. This is experimental and might be removed in a future release.
advi: Run ADVI to estimate posterior mean and diagonal mass matrix.
advi_map: Initialize ADVI with MAP and use MAP as starting point.
map: Use the MAP as starting point. This is discouraged.
adapt_full: Adapt a dense mass matrix using the sample covariances

stepfunction or iterable of functions

A step function or collection of functions. If there are variables without step methods, step methods for those variables will be assigned automatically. By default the NUTS step method will be used, if appropriate to the model; this is a good default for beginning users.

n_initint

Number of iterations of initializer. Only works for ‘ADVI’ init methods.

startdict, or array of dict

Starting point in parameter space (or partial point) Defaults to trace.point(-1)) if there is a trace provided and model.test_point if not (defaults to empty dict). Initialization methods for NUTS (see init keyword) can overwrite the default.

tracebackend, list, or MultiTrace

This should be a backend instance, a list of variables to track, or a MultiTrace object with past values. If a MultiTrace object is given, it must contain samples for the chain number chain. If None or a list of variables, the NDArray backend is used.

chain_idxint

Chain number used to store sample in backend. If chains is greater than one, chain numbers will start here.

chainsint

The number of chains to sample. Running independent chains is important for some convergence statistics and can also reveal multiple modes in the posterior. If None, then set to either cores or 2, whichever is larger.

coresint

The number of chains to run in parallel. If None, set to the number of CPUs in the system, but at most 4.

tuneint

Number of iterations to tune, defaults to 1000. Samplers adjust the step sizes, scalings or similar during tuning. Tuning samples will be drawn in addition to the number specified in the draws argument, and will be discarded unless discard_tuned_samples is set to False.

progressbarbool, optional default=True

Whether or not to display a progress bar in the command line. The bar shows the percentage of completion, the sampling speed in samples per second (SPS), and the estimated remaining time until completion (“expected time of arrival”; ETA).

modelModel (optional if in with context)

random_seedint or list of ints

A list is accepted if cores is greater than one.

discard_tuned_samplesbool

Whether to discard posterior samples of the tune interval.

compute_convergence_checksbool, default=True

Whether to compute sampler statistics like Gelman-Rubin and effective_n.

callbackfunction, default=None

A function which gets called for every sample from the trace of a chain. The function is called with the trace and the current draw and will contain all samples for a single trace. the draw.chain argument can be used to determine which of the active chains the sample is drawn from. Sampling can be interrupted by throwing a KeyboardInterrupt in the callback.

jitter_max_retriesint

return_inferencedatabool, default=False

Whether to return the trace as an arviz.InferenceData (True) object or a MultiTrace (False) Defaults to False, but we’ll switch to True in an upcoming release.

idata_kwargsdict, optional

Keyword arguments for arviz.from_pymc3()

mp_ctxmultiprocessing.context.BaseContent

A multiprocessing context for parallel sampling. See multiprocessing documentation for details.

pickle_backendstr

One of ‘pickle’ or ‘dill’. The library used to pickle models in parallel sampling if the multiprocessing context is not of type fork.

Returns

tracepymc3.backends.base.MultiTrace or arviz.InferenceData: A MultiTrace or ArviZ InferenceData object that contains the samples.

Notes

Optional keyword arguments can be passed to sample to be delivered to the step_methods used during sampling.

If your model uses only one step method, you can address step method kwargs directly. In particular, the NUTS step method has several options including:

target_accept : float in [0, 1]. The step size is tuned such that we approximate this acceptance rate. Higher values like 0.9 or 0.95 often work better for problematic posteriors

max_treedepth : The maximum depth of the trajectory tree

step_scale : float, default 0.25 The initial guess for the step size scaled down by \(1/n**(1/4)\)

If your model uses multiple step methods, aka a Compound Step, then you have two ways to address arguments to each step method:

If you let sample() automatically assign the step_methods, and you can correctly anticipate what they will be, then you can wrap step method kwargs in a dict and pass that to sample() with a kwarg set to the name of the step method. e.g. for a CompoundStep comprising NUTS and BinaryGibbsMetropolis, you could send:
1. target_accept to NUTS: nuts={‘target_accept’:0.9}
2. transit_p to BinaryGibbsMetropolis: binary_gibbs_metropolis={‘transit_p’:.7}
Note that available names are:

nuts, hmc, metropolis, binary_metropolis, binary_gibbs_metropolis, categorical_gibbs_metropolis, DEMetropolis, DEMetropolisZ, slice
If you manually declare the step_methods, within the step kwarg, then you can address the step_method kwargs directly. e.g. for a CompoundStep comprising NUTS and BinaryGibbsMetropolis, you could send
```
step=[pm.NUTS([freeRV1, freeRV2], target_accept=0.9),
      pm.BinaryGibbsMetropolis([freeRV3], transit_p=.7)]
```

You can find a full list of arguments in the docstring of the step methods.

Examples

In [1]: import pymc3 as pm
   ...: n = 100
   ...: h = 61
   ...: alpha = 2
   ...: beta = 2

In [2]: with pm.Model() as model: # context management
   ...:     p = pm.Beta("p", alpha=alpha, beta=beta)
   ...:     y = pm.Binomial("y", n=n, p=p, observed=h)
   ...:     trace = pm.sample()

In [3]: az.summary(trace, kind="stats")

Out[3]:
    mean     sd  hdi_3%  hdi_97%
p  0.609  0.047   0.528    0.699

pymc3.sampling.sample_posterior_predictive(trace, samples: Optional[int] = None, model: Optional[pymc3.model.Model] = None, var_names: Optional[List[str]] = None, size: Optional[int] = None, keep_size: Optional[bool] = False, random_seed=None, progressbar: bool = True) → Dict[str, numpy.ndarray]¶

Generate posterior predictive samples from a model given a trace.

Parameters

tracebackend, list, xarray.Dataset, arviz.InferenceData, or MultiTrace: Trace generated from MCMC sampling, or a list of dicts (eg. points or from find_MAP()), or xarray.Dataset (eg. InferenceData.posterior or InferenceData.prior)
samplesint: Number of posterior predictive samples to generate. Defaults to one posterior predictive sample per posterior sample, that is, the number of draws times the number of chains. It is not recommended to modify this value; when modified, some chains may not be represented in the posterior predictive sample.
modelModel (optional if in with context): Model used to generate trace
varsiterable: Variables for which to compute the posterior predictive samples. Deprecated: please use var_names instead.
var_namesIterable[str]: Names of variables for which to compute the posterior predictive samples.
sizeint: The number of random draws from the distribution specified by the parameters in each sample of the trace. Not recommended unless more than ndraws times nchains posterior predictive samples are needed.
keep_sizebool, optional: Force posterior predictive sample to have the same shape as posterior and sample stats data: (nchains, ndraws, ...). Overrides samples and size parameters.
random_seedint: Seed for the random number generator.
progressbarbool: Whether or not to display a progress bar in the command line. The bar shows the percentage of completion, the sampling speed in samples per second (SPS), and the estimated remaining time until completion (“expected time of arrival”; ETA).

Returns

samplesdict: Dictionary with the variable names as keys, and values numpy arrays containing posterior predictive samples.

pymc3.sampling.sample_posterior_predictive_w(traces, samples: Optional[int] = None, models: Optional[List[pymc3.model.Model]] = None, weights: Optional[Union[numpy.ndarray, List[float]]] = None, random_seed: Optional[int] = None, progressbar: bool = True)¶

Generate weighted posterior predictive samples from a list of models and a list of traces according to a set of weights.

Parameters

traceslist or list of lists: List of traces generated from MCMC sampling (xarray.Dataset, arviz.InferenceData, or MultiTrace), or a list of list containing dicts from find_MAP() or points. The number of traces should be equal to the number of weights.
samplesint, optional: Number of posterior predictive samples to generate. Defaults to the length of the shorter trace in traces.
modelslist of Model: List of models used to generate the list of traces. The number of models should be equal to the number of weights and the number of observed RVs should be the same for all models. By default a single model will be inferred from with context, in this case results will only be meaningful if all models share the same distributions for the observed RVs.
weightsarray-like, optional: Individual weights for each trace. Default, same weight for each model.
random_seedint, optional: Seed for the random number generator.
progressbarbool, optional default True: Whether or not to display a progress bar in the command line. The bar shows the percentage of completion, the sampling speed in samples per second (SPS), and the estimated remaining time until completion (“expected time of arrival”; ETA).

Returns

samplesdict: Dictionary with the variables as keys. The values corresponding to the posterior predictive samples from the weighted models.

pymc3.sampling.sample_prior_predictive(samples=500, model: Optional[pymc3.model.Model] = None, var_names: Optional[Iterable[str]] = None, random_seed=None) → Dict[str, numpy.ndarray]¶

Generate samples from the prior predictive distribution.

Parameters

samplesint: Number of samples from the prior predictive to generate. Defaults to 500.
modelModel (optional if in with context)
var_namesIterable[str]: A list of names of variables for which to compute the posterior predictive samples. Defaults to both observed and unobserved RVs.
random_seedint: Seed for the random number generator.

Returns

dict: Dictionary with variable names as keys. The values are numpy arrays of prior samples.

Step-methods¶

pymc3.sampling.assign_step_methods(model, step=None, methods=(<class 'pymc3.step_methods.hmc.nuts.NUTS'>, <class 'pymc3.step_methods.hmc.hmc.HamiltonianMC'>, <class 'pymc3.step_methods.metropolis.Metropolis'>, <class 'pymc3.step_methods.metropolis.BinaryMetropolis'>, <class 'pymc3.step_methods.metropolis.BinaryGibbsMetropolis'>, <class 'pymc3.step_methods.slicer.Slice'>, <class 'pymc3.step_methods.metropolis.CategoricalGibbsMetropolis'>, <class 'pymc3.step_methods.pgbart.PGBART'>), step_kwargs=None)¶

Assign model variables to appropriate step methods.

Passing a specified model will auto-assign its constituent stochastic variables to step methods based on the characteristics of the variables. This function is intended to be called automatically from sample(), but may be called manually. Each step method passed should have a competence() method that returns an ordinal competence value corresponding to the variable passed to it. This value quantifies the appropriateness of the step method for sampling the variable.

Parameters

modelModel object: A fully-specified model object
stepstep function or vector of step functions: One or more step functions that have been assigned to some subset of the model’s parameters. Defaults to None (no assigned variables).
methodsvector of step method classes: The set of step methods from which the function may choose. Defaults to the main step methods provided by PyMC3.
step_kwargsdict: Parameters for the samplers. Keys are the lower case names of the step method, values a dict of arguments.

Returns

methodslist: List of step methods associated with the model’s variables.

NUTS¶

class pymc3.step_methods.hmc.nuts.NUTS(*args, **kwargs)¶

A sampler for continuous variables based on Hamiltonian mechanics.

NUTS automatically tunes the step size and the number of steps per sample. A detailed description can be found at [1], “Algorithm 6: Efficient No-U-Turn Sampler with Dual Averaging”.

NUTS provides a number of statistics that can be accessed with trace.get_sampler_stats:

mean_tree_accept: The mean acceptance probability for the tree that generated this sample. The mean of these values across all samples but the burn-in should be approximately target_accept (the default for this is 0.8).
diverging: Whether the trajectory for this sample diverged. If there are any divergences after burnin, this indicates that the results might not be reliable. Reparametrization can often help, but you can also try to increase target_accept to something like 0.9 or 0.95.
energy: The energy at the point in phase-space where the sample was accepted. This can be used to identify posteriors with problematically long tails. See below for an example.
energy_change: The difference in energy between the start and the end of the trajectory. For a perfect integrator this would always be zero.
max_energy_change: The maximum difference in energy along the whole trajectory.
depth: The depth of the tree that was used to generate this sample
tree_size: The number of leafs of the sampling tree, when the sample was accepted. This is usually a bit less than 2 ** depth. If the tree size is large, the sampler is using a lot of leapfrog steps to find the next sample. This can for example happen if there are strong correlations in the posterior, if the posterior has long tails, if there are regions of high curvature (“funnels”), or if the variance estimates in the mass matrix are inaccurate. Reparametrisation of the model or estimating the posterior variances from past samples might help.
tune: This is True, if step size adaptation was turned on when this sample was generated.
step_size: The step size used for this sample.
step_size_bar: The current best known step-size. After the tuning samples, the step size is set to this value. This should converge during tuning.
model_logp: The model log-likelihood for this sample.
process_time_diff: The time it took to draw the sample, as defined by the python standard library time.process_time. This counts all the CPU time, including worker processes in BLAS and OpenMP.
perf_counter_diff: The time it took to draw the sample, as defined by the python standard library time.perf_counter (wall time).
perf_counter_start: The value of time.perf_counter at the beginning of the computation of the draw.

References

1: Hoffman, Matthew D., & Gelman, Andrew. (2011). The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo.

Set up the No-U-Turn sampler.

Parameters

vars: list of Theano variables, default all continuous vars
Emax: float, default 1000: Maximum energy change allowed during leapfrog steps. Larger deviations will abort the integration.
target_accept: float, default .8: Adapt the step size such that the average acceptance probability across the trajectories are close to target_accept. Higher values for target_accept lead to smaller step sizes. Setting this to higher values like 0.9 or 0.99 can help with sampling from difficult posteriors. Valid values are between 0 and 1 (exclusive).
step_scale: float, default 0.25: Size of steps to take, automatically scaled down by 1/n**(1/4). If step size adaptation is switched off, the resulting step size is used. If adaptation is enabled, it is used as initial guess.
gamma: float, default .05
k: float, default .75: Parameter for dual averaging for step size adaptation. Values between 0.5 and 1 (exclusive) are admissible. Higher values correspond to slower adaptation.
t0: int, default 10: Parameter for dual averaging. Higher values slow initial adaptation.
adapt_step_size: bool, default=True: Whether step size adaptation should be enabled. If this is disabled, k, t0, gamma and target_accept are ignored.
max_treedepth: int, default=10: The maximum tree depth. Trajectories are stopped when this depth is reached.
early_max_treedepth: int, default=8: The maximum tree depth during the first 200 tuning samples.
scaling: array_like, ndim = {1,2}: The inverse mass, or precision matrix. One dimensional arrays are interpreted as diagonal matrices. If is_cov is set to True, this will be interpreded as the mass or covariance matrix.
is_cov: bool, default=False: Treat the scaling as mass or covariance matrix.
potential: Potential, optional: An object that represents the Hamiltonian with methods velocity, energy, and random methods. It can be specified instead of the scaling matrix.
model: pymc3.Model: The model
kwargs: passed to BaseHMC

Notes

The step size adaptation stops when self.tune is set to False. This is usually achieved by setting the tune parameter if pm.sample to the desired number of tuning steps.

static competence(var, has_grad)¶: Check how appropriate this class is for sampling a random variable.

Metropolis¶

class pymc3.step_methods.metropolis.BinaryGibbsMetropolis(*args, **kwargs)¶

A Metropolis-within-Gibbs step method optimized for binary variables

Parameters

vars: list: List of variables for sampler
order: list or ‘random’: List of integers indicating the Gibbs update order e.g., [0, 2, 1, …]. Default is random
transit_p: float: The diagonal of the transition kernel. A value > .5 gives anticorrelated proposals, which resulting in more efficient antithetical sampling. Default is 0.8
model: PyMC Model: Optional model for sampling step. Defaults to None (taken from context).

static competence(var)¶: BinaryMetropolis is only suitable for Bernoulli and Categorical variables with k=2.

class pymc3.step_methods.metropolis.BinaryMetropolis(*args, **kwargs)¶

Metropolis-Hastings optimized for binary variables

Parameters

vars: list: List of variables for sampler
scaling: scalar or array: Initial scale factor for proposal. Defaults to 1.
tune: bool: Flag for tuning. Defaults to True.
tune_interval: int: The frequency of tuning. Defaults to 100 iterations.
model: PyMC Model: Optional model for sampling step. Defaults to None (taken from context).

static competence(var)¶: BinaryMetropolis is only suitable for binary (bool) and Categorical variables with k=1.

class pymc3.step_methods.metropolis.CategoricalGibbsMetropolis(*args, **kwargs)¶

A Metropolis-within-Gibbs step method optimized for categorical variables. This step method works for Bernoulli variables as well, but it is not optimized for them, like BinaryGibbsMetropolis is. Step method supports two types of proposals: A uniform proposal and a proportional proposal, which was introduced by Liu in his 1996 technical report “Metropolized Gibbs Sampler: An Improvement”.

static competence(var)¶: CategoricalGibbsMetropolis is only suitable for Bernoulli and Categorical variables.

class pymc3.step_methods.metropolis.DEMetropolis(*args, **kwargs)¶

Differential Evolution Metropolis sampling step.

Parameters

lamb: float: Lambda parameter of the DE proposal mechanism. Defaults to 2.38 / sqrt(2 * ndim)
vars: list: List of variables for sampler
S: standard deviation or covariance matrix: Some measure of variance to parameterize proposal distribution
proposal_dist: function: Function that returns zero-mean deviates when parameterized with S (and n). Defaults to Uniform(-S,+S).
scaling: scalar or array: Initial scale factor for epsilon. Defaults to 0.001
tune: str: Which hyperparameter to tune. Defaults to None, but can also be ‘scaling’ or ‘lambda’.
tune_interval: int: The frequency of tuning. Defaults to 100 iterations.
model: PyMC Model: Optional model for sampling step. Defaults to None (taken from context).
mode: string or `Mode` instance.: compilation mode passed to Theano functions

References

Braak2006: Cajo C.F. ter Braak (2006). A Markov Chain Monte Carlo version of the genetic algorithm Differential Evolution: easy Bayesian computing for real parameter spaces. Statistics and Computing link

Parameters

vars: list of sampling variables
shared: dict of theano variable -> shared variable
blocked: Boolean (default True)

class pymc3.step_methods.metropolis.DEMetropolisZ(*args, **kwargs)¶

Adaptive Differential Evolution Metropolis sampling step that uses the past to inform jumps.

Parameters

lamb: float: Lambda parameter of the DE proposal mechanism. Defaults to 2.38 / sqrt(2 * ndim)
vars: list: List of variables for sampler
S: standard deviation or covariance matrix: Some measure of variance to parameterize proposal distribution
proposal_dist: function: Function that returns zero-mean deviates when parameterized with S (and n). Defaults to Uniform(-S,+S).
scaling: scalar or array: Initial scale factor for epsilon. Defaults to 0.001
tune: str: Which hyperparameter to tune. Defaults to ‘lambda’, but can also be ‘scaling’ or None.
tune_interval: int: The frequency of tuning. Defaults to 100 iterations.
tune_drop_fraction: float: Fraction of tuning steps that will be removed from the samplers history when the tuning ends. Defaults to 0.9 - keeping the last 10% of tuning steps for good mixing while removing 90% of potentially unconverged tuning positions.
model: PyMC Model: Optional model for sampling step. Defaults to None (taken from context).
mode: string or `Mode` instance.: compilation mode passed to Theano functions

References

Braak2006: Cajo C.F. ter Braak (2006). Differential Evolution Markov Chain with snooker updater and fewer chains. Statistics and Computing link

Parameters

vars: list of sampling variables
shared: dict of theano variable -> shared variable
blocked: Boolean (default True)

reset_tuning()¶: Resets the tuned sampler parameters and history to their initial values.

stop_tuning()¶: At the end of the tuning phase, this method removes the first x% of the history so future proposals are not informed by unconverged tuning iterations.

class pymc3.step_methods.metropolis.Metropolis(*args, **kwargs)¶

Metropolis-Hastings sampling step

Create an instance of a Metropolis stepper

Parameters

vars: list: List of variables for sampler
S: standard deviation or covariance matrix: Some measure of variance to parameterize proposal distribution
proposal_dist: function: Function that returns zero-mean deviates when parameterized with S (and n). Defaults to normal.
scaling: scalar or array: Initial scale factor for proposal. Defaults to 1.
tune: bool: Flag for tuning. Defaults to True.
tune_interval: int: The frequency of tuning. Defaults to 100 iterations.
model: PyMC Model: Optional model for sampling step. Defaults to None (taken from context).
mode: string or `Mode` instance.: compilation mode passed to Theano functions

reset_tuning()¶: Resets the tuned sampler parameters to their initial values.

Slice¶

class pymc3.step_methods.slicer.Slice(*args, **kwargs)¶

Univariate slice sampler step method

Parameters

vars: list: List of variables for sampler.
w: float: Initial width of slice (Defaults to 1).
tune: bool: Flag for tuning (Defaults to True).
model: PyMC Model: Optional model for sampling step. Defaults to None (taken from context).

Hamiltonian Monte Carlo¶

class pymc3.step_methods.hmc.hmc.HamiltonianMC(*args, **kwargs)¶

A sampler for continuous variables based on Hamiltonian mechanics.

See NUTS sampler for automatically tuned stopping time and step size scaling.

Set up the Hamiltonian Monte Carlo sampler.

Parameters

vars: list of theano variables
path_length: float, default=2: total length to travel
step_rand: function float -> float, default=unif: A function which takes the step size and returns an new one used to randomize the step size at each iteration.
step_scale: float, default=0.25: Initial size of steps to take, automatically scaled down by 1/n**(1/4).
scaling: array_like, ndim = {1,2}: The inverse mass, or precision matrix. One dimensional arrays are interpreted as diagonal matrices. If is_cov is set to True, this will be interpreded as the mass or covariance matrix.
is_cov: bool, default=False: Treat the scaling as mass or covariance matrix.
potential: Potential, optional: An object that represents the Hamiltonian with methods velocity, energy, and random methods. It can be specified instead of the scaling matrix.
target_accept: float, default 0.65: Adapt the step size such that the average acceptance probability across the trajectories are close to target_accept. Higher values for target_accept lead to smaller step sizes. Setting this to higher values like 0.9 or 0.99 can help with sampling from difficult posteriors. Valid values are between 0 and 1 (exclusive). Default of 0.65 is from (Beskos et. al. 2010, Neal 2011). See Hoffman and Gelman’s “The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo” section 3.2 for details.
gamma: float, default .05
k: float, default .75: Parameter for dual averaging for step size adaptation. Values between 0.5 and 1 (exclusive) are admissible. Higher values correspond to slower adaptation.
t0: float > 0, default 10: Parameter for dual averaging. Higher values slow initial adaptation.
adapt_step_size: bool, default=True: Whether step size adaptation should be enabled. If this is disabled, k, t0, gamma and target_accept are ignored.
max_steps: int: The maximum number of leapfrog steps.
model: pymc3.Model: The model
**kwargs: passed to BaseHMC

static competence(var, has_grad)¶: Check how appropriate this class is for sampling a random variable.

Sequential Monte Carlo¶

class pymc3.smc.smc.SMC(draws=2000, kernel='metropolis', n_steps=25, start=None, tune_steps=True, p_acc_rate=0.85, threshold=0.5, save_sim_data=False, save_log_pseudolikelihood=True, model=None, random_seed=- 1, chain=0)¶

Sequential Monte Carlo with Independent Metropolis-Hastings and ABC kernels.

initialize_logp()¶: Initialize the prior and likelihood log probabilities.

initialize_population()¶: Create an initial population from the prior distribution.

mutate()¶: Independent Metropolis-Hastings perturbation.

posterior_to_trace()¶: Save results into a PyMC3 trace.

resample()¶: Resample particles based on importance weights.

setup_kernel()¶: Set up the likelihood logp function based on the chosen kernel.

tune()¶: Tune n_steps based on the acceptance rate.

update_proposal()¶: Update proposal based on the covariance matrix from tempered posterior.

update_weights_beta()¶

Calculate the next inverse temperature (beta).

The importance weights based on current beta and tempered likelihood and updates the marginal likelihood estimate.

MultiTrace¶

class pymc3.backends.base.MultiTrace(straces)¶

Main interface for accessing values from MCMC results.

The core method to select values is get_values. The method to select sampler statistics is get_sampler_stats. Both kinds of values can also be accessed by indexing the MultiTrace object. Indexing can behave in four ways:

Indexing with a variable or variable name (str) returns all values for that variable, combining values for all chains.
```
>>> trace[varname]
```
Slicing after the variable name can be used to burn and thin the samples.
```
>>> trace[varname, 1000:]
```
For convenience during interactive use, values can also be accessed using the variable as an attribute.
```
>>> trace.varname
```
Indexing with an integer returns a dictionary with values for each variable at the given index (corresponding to a single sampling iteration).
Slicing with a range returns a new trace with the number of draws corresponding to the range.
Indexing with the name of a sampler statistic that is not also the name of a variable returns those values from all chains. If there is more than one sampler that provides that statistic, the values are concatenated along a new axis.

For any methods that require a single trace (e.g., taking the length of the MultiTrace instance, which returns the number of draws), the trace with the highest chain number is always used.

Attributes

nchains: int: Number of chains in the MultiTrace.
chains: `List[int]`: List of chain indices
report: str: Report on the sampling process.
varnames: `List[str]`: List of variable names in the trace(s)

add_values(vals, overwrite=False) → None¶

Add variables to traces.

Parameters

vals: dict (str: array-like): The keys should be the names of the new variables. The values are expected to be array-like objects. For traces with more than one chain the length of each value should match the number of total samples already in the trace (chains * iterations), otherwise a warning is raised.
overwrite: bool: If False (default) a ValueError is raised if the variable already exists. Change to True to overwrite the values of variables

Returns

None.

get_sampler_stats(stat_name, burn=0, thin=1, combine=True, chains=None, squeeze=True)¶

Get sampler statistics from the trace.

Parameters

stat_name: str
sampler_idx: int or None
burn: int
thin: int

Returns

If the sampler_idx is specified, return the statistic with
the given name in a numpy array. If it is not specified and there
is more than one sampler that provides this statistic, return
a numpy array of shape (m, n), where m is the number of
such samplers, and n is the number of samples.

get_values(varname, burn=0, thin=1, combine=True, chains=None, squeeze=True)¶

Get values from traces.

Parameters

varname: str
burn: int
thin: int
combine: bool: If True, results from chains will be concatenated.
chains: int or list of ints: Chains to retrieve. If None, all chains are used. A single chain value can also be given.
squeeze: bool: Return a single array element if the resulting list of values only has one element. If False, the result will always be a list of arrays, even if combine is True.

Returns

A list of NumPy arrays or a single NumPy array (depending on
squeeze).

point(idx, chain=None)¶

Return a dictionary of point values at idx.

Parameters

idx: int
chain: int: If a chain is not given, the highest chain number is used.

points(chains=None)¶

Return an iterator over all or some of the sample points

Parameters

chains: list of int or N: The chains whose points should be inlcuded in the iterator. If chains is not given, include points from all chains.

remove_values(name)¶

remove variables from traces.

Parameters

name: str: Name of the variable to remove. Raises KeyError if the variable is not present

class pymc3.backends.base.BaseTrace(name, model=None, vars=None, test_point=None)¶

Base trace object

Parameters

name: str: Name of backend
model: Model: If None, the model is taken from the with context.
vars: list of variables: Sampling values will be stored for these variables. If None, model.unobserved_RVs is used.
test_point: dict: use different test point that might be with changed variables shapes

Variational Inference¶

OPVI¶

Variational inference is a great approach for doing really complex, often intractable Bayesian inference in approximate form. Common methods (e.g. ADVI) lack from complexity so that approximate posterior does not reveal the true nature of underlying problem. In some applications it can yield unreliable decisions.

Recently on NIPS 2017 OPVI framework was presented. It generalizes variational inference so that the problem is build with blocks. The first and essential block is Model itself. Second is Approximation, in some cases \(log Q(D)\) is not really needed. Necessity depends on the third and fourth part of that black box, Operator and Test Function respectively.

Operator is like an approach we use, it constructs loss from given Model, Approximation and Test Function. The last one is not needed if we minimize KL Divergence from Q to posterior. As a drawback we need to compute \(loq Q(D)\). Sometimes approximation family is intractable and \(loq Q(D)\) is not available, here comes LS(Langevin Stein) Operator with a set of test functions.

Test Function has more unintuitive meaning. It is usually used with LS operator and represents all we want from our approximate distribution. For any given vector based function of \(z\) LS operator yields zero mean function under posterior. \(loq Q(D)\) is no more needed. That opens a door to rich approximation families as neural networks.

References

Rajesh Ranganath, Jaan Altosaar, Dustin Tran, David M. Blei Operator Variational Inference https://arxiv.org/abs/1610.09033 (2016)

class pymc3.variational.opvi.Approximation(groups, model=None)¶

Wrapper for grouped approximations

Wraps list of groups, creates an Approximation instance that collects sampled variables from all the groups, also collects logQ needed for explicit Variational Inference.

Parameters

groups: list[Group]: List of Group instances. They should have all model variables
model: Model

VI Inference API¶

class pymc3.variational.inference.ADVI(*args, **kwargs)¶

Automatic Differentiation Variational Inference (ADVI)

This class implements the meanfield ADVI, where the variational posterior distribution is assumed to be spherical Gaussian without correlation of parameters and fit to the true posterior distribution. The means and standard deviations of the variational posterior are referred to as variational parameters.

For explanation, we classify random variables in probabilistic models into three types. Observed random variables \({\cal Y}=\{\mathbf{y}_{i}\}_{i=1}^{N}\) are \(N\) observations. Each \(\mathbf{y}_{i}\) can be a set of observed random variables, i.e., \(\mathbf{y}_{i}=\{\mathbf{y}_{i}^{k}\}_{k=1}^{V_{o}}\), where \(V_{k}\) is the number of the types of observed random variables in the model.

The next ones are global random variables \(\Theta=\{\theta^{k}\}_{k=1}^{V_{g}}\), which are used to calculate the probabilities for all observed samples.

The last ones are local random variables \({\cal Z}=\{\mathbf{z}_{i}\}_{i=1}^{N}\), where \(\mathbf{z}_{i}=\{\mathbf{z}_{i}^{k}\}_{k=1}^{V_{l}}\). These RVs are used only in AEVB.

The goal of ADVI is to approximate the posterior distribution \(p(\Theta,{\cal Z}|{\cal Y})\) by variational posterior \(q(\Theta)\prod_{i=1}^{N}q(\mathbf{z}_{i})\). All of these terms are normal distributions (mean-field approximation).

\(q(\Theta)\) is parametrized with its means and standard deviations. These parameters are denoted as \(\gamma\). While \(\gamma\) is a constant, the parameters of \(q(\mathbf{z}_{i})\) are dependent on each observation. Therefore these parameters are denoted as \(\xi(\mathbf{y}_{i}; \nu)\), where \(\nu\) is the parameters of \(\xi(\cdot)\). For example, \(\xi(\cdot)\) can be a multilayer perceptron or convolutional neural network.

In addition to \(\xi(\cdot)\), we can also include deterministic mappings for the likelihood of observations. We denote the parameters of the deterministic mappings as \(\eta\). An example of such mappings is the deconvolutional neural network used in the convolutional VAE example in the PyMC3 notebook directory.

This function maximizes the evidence lower bound (ELBO) \({\cal L}(\gamma, \nu, \eta)\) defined as follows:

\[\begin{split}{\cal L}(\gamma,\nu,\eta) & = \mathbf{c}_{o}\mathbb{E}_{q(\Theta)}\left[ \sum_{i=1}^{N}\mathbb{E}_{q(\mathbf{z}_{i})}\left[ \log p(\mathbf{y}_{i}|\mathbf{z}_{i},\Theta,\eta) \right]\right] \\ & - \mathbf{c}_{g}KL\left[q(\Theta)||p(\Theta)\right] - \mathbf{c}_{l}\sum_{i=1}^{N} KL\left[q(\mathbf{z}_{i})||p(\mathbf{z}_{i})\right],\end{split}\]

where \(KL[q(v)||p(v)]\) is the Kullback-Leibler divergence

\[KL[q(v)||p(v)] = \int q(v)\log\frac{q(v)}{p(v)}dv,\]

\(\mathbf{c}_{o/g/l}\) are vectors for weighting each term of ELBO. More precisely, we can write each of the terms in ELBO as follows:

\[\begin{split}\mathbf{c}_{o}\log p(\mathbf{y}_{i}|\mathbf{z}_{i},\Theta,\eta) & = & \sum_{k=1}^{V_{o}}c_{o}^{k} \log p(\mathbf{y}_{i}^{k}| {\rm pa}(\mathbf{y}_{i}^{k},\Theta,\eta)) \\ \mathbf{c}_{g}KL\left[q(\Theta)||p(\Theta)\right] & = & \sum_{k=1}^{V_{g}}c_{g}^{k}KL\left[ q(\theta^{k})||p(\theta^{k}|{\rm pa(\theta^{k})})\right] \\ \mathbf{c}_{l}KL\left[q(\mathbf{z}_{i}||p(\mathbf{z}_{i})\right] & = & \sum_{k=1}^{V_{l}}c_{l}^{k}KL\left[ q(\mathbf{z}_{i}^{k})|| p(\mathbf{z}_{i}^{k}|{\rm pa}(\mathbf{z}_{i}^{k}))\right],\end{split}\]

where \({\rm pa}(v)\) denotes the set of parent variables of \(v\) in the directed acyclic graph of the model.

When using mini-batches, \(c_{o}^{k}\) and \(c_{l}^{k}\) should be set to \(N/M\), where \(M\) is the number of observations in each mini-batch. This is done with supplying total_size parameter to observed nodes (e.g. Normal('x', 0, 1, observed=data, total_size=10000)). In this case it is possible to automatically determine appropriate scaling for \(logp\) of observed nodes. Interesting to note that it is possible to have two independent observed variables with different total_size and iterate them independently during inference.

For working with ADVI, we need to give

The probabilistic model

model with three types of RVs (observed_RVs, global_RVs and local_RVs).
(optional) Minibatches

The tensors to which mini-bathced samples are supplied are handled separately by using callbacks in Inference.fit() method that change storage of shared theano variable or by pymc3.generator() that automatically iterates over minibatches and defined beforehand.
(optional) Parameters of deterministic mappings

They have to be passed along with other params to Inference.fit() method as more_obj_params argument.

For more information concerning training stage please reference pymc3.variational.opvi.ObjectiveFunction.step_function()

Parameters

local_rv: dict[var->tuple]: mapping {model_variable -> approx params} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details
model: :class:`pymc3.Model`: PyMC3 model for inference
random_seed: None or int: leave None to use package global RandomStream or other valid value to create instance specific one
start: `Point`: starting point for inference

References

Kucukelbir, A., Tran, D., Ranganath, R., Gelman, A., and Blei, D. M. (2016). Automatic Differentiation Variational Inference. arXiv preprint arXiv:1603.00788.
Geoffrey Roeder, Yuhuai Wu, David Duvenaud, 2016 Sticking the Landing: A Simple Reduced-Variance Gradient for ADVI approximateinference.org/accepted/RoederEtAl2016.pdf
Kingma, D. P., & Welling, M. (2014). Auto-Encoding Variational Bayes. stat, 1050, 1.

class pymc3.variational.inference.ASVGD(approx=None, estimator=<class 'pymc3.variational.operators.KSD'>, kernel=<pymc3.variational.test_functions.RBF object>, **kwargs)¶

Amortized Stein Variational Gradient Descent

not suggested to use

This inference is based on Kernelized Stein Discrepancy it’s main idea is to move initial noisy particles so that they fit target distribution best.

Algorithm is outlined below

Input: Parametrized random generator \(R_{\theta}\)

Output: \(R_{\theta^{*}}\) that approximates the target distribution.

\[\begin{split}\Delta x_i &= \hat{\phi}^{*}(x_i) \\ \hat{\phi}^{*}(x) &= \frac{1}{n}\sum^{n}_{j=1}[k(x_j,x) \nabla_{x_j} logp(x_j)+ \nabla_{x_j} k(x_j,x)] \\ \Delta_{\theta} &= \frac{1}{n}\sum^{n}_{i=1}\Delta x_i\frac{\partial x_i}{\partial \theta}\end{split}\]

Parameters

approx: :class:`Approximation`: default is FullRank but can be any
kernel: `callable`: kernel function for KSD \(f(histogram) -> (k(x,.), \nabla_x k(x,.))\)
model: :class:`Model`
kwargs: kwargs for gradient estimator

References

Dilin Wang, Yihao Feng, Qiang Liu (2016) Learning to Sample Using Stein Discrepancy http://bayesiandeeplearning.org/papers/BDL_21.pdf
Dilin Wang, Qiang Liu (2016) Learning to Draw Samples: With Application to Amortized MLE for Generative Adversarial Learning arXiv:1611.01722
Yang Liu, Prajit Ramachandran, Qiang Liu, Jian Peng (2017) Stein Variational Policy Gradient arXiv:1704.02399

fit(n=10000, score=None, callbacks=None, progressbar=True, obj_n_mc=500, **kwargs)¶

Perform Operator Variational Inference

Parameters

n: int: number of iterations
score: bool: evaluate loss on each iteration or not
callbacks: list[function: (Approximation, losses, i) -> None]: calls provided functions after each iteration step
progressbar: bool: whether to show progressbar or not

Returns

Approximation

Other Parameters

obj_n_mc: `int`: Number of monte carlo samples used for approximation of objective gradients
tf_n_mc: `int`: Number of monte carlo samples used for approximation of test function gradients
obj_optimizer: function (grads, params) -> updates: Optimizer that is used for objective params
test_optimizer: function (grads, params) -> updates: Optimizer that is used for test function params
more_obj_params: `list`: Add custom params for objective optimizer
more_tf_params: `list`: Add custom params for test function optimizer
more_updates: `dict`: Add custom updates to resulting updates
total_grad_norm_constraint: `float`: Bounds gradient norm, prevents exploding gradient problem
fn_kwargs: `dict`: Add kwargs to theano.function (e.g. {‘profile’: True})
more_replacements: `dict`: Apply custom replacements before calculating gradients

class pymc3.variational.inference.FullRankADVI(*args, **kwargs)¶

Full Rank Automatic Differentiation Variational Inference (ADVI)

Parameters

local_rv: dict[var->tuple]: mapping {model_variable -> approx params} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details
model: :class:`pymc3.Model`: PyMC3 model for inference
random_seed: None or int: leave None to use package global RandomStream or other valid value to create instance specific one
start: `Point`: starting point for inference

References

Kucukelbir, A., Tran, D., Ranganath, R., Gelman, A., and Blei, D. M. (2016). Automatic Differentiation Variational Inference. arXiv preprint arXiv:1603.00788.
Geoffrey Roeder, Yuhuai Wu, David Duvenaud, 2016 Sticking the Landing: A Simple Reduced-Variance Gradient for ADVI approximateinference.org/accepted/RoederEtAl2016.pdf
Kingma, D. P., & Welling, M. (2014). Auto-Encoding Variational Bayes. stat, 1050, 1.

class pymc3.variational.inference.ImplicitGradient(approx, estimator=<class 'pymc3.variational.operators.KSD'>, kernel=<pymc3.variational.test_functions.RBF object>, **kwargs)¶

Implicit Gradient for Variational Inference

not suggested to use

An approach to fit arbitrary approximation by computing kernel based gradient By default RBF kernel is used for gradient estimation. Default estimator is Kernelized Stein Discrepancy with temperature equal to 1. This temperature works only for large number of samples. Larger temperature is needed for small number of samples but there is no theoretical approach to choose the best one in such case.

class pymc3.variational.inference.Inference(op, approx, tf, **kwargs)¶

Base class for Variational Inference

Communicates Operator, Approximation and Test Function to build Objective Function

Parameters

op: Operator class
approx: Approximation class or instance
tf: TestFunction instance
model: Model: PyMC3 Model
kwargs: kwargs passed to :class:`Operator`

fit(n=10000, score=None, callbacks=None, progressbar=True, **kwargs)¶

Perform Operator Variational Inference

Parameters

n: int: number of iterations
score: bool: evaluate loss on each iteration or not
callbacks: list[function: (Approximation, losses, i) -> None]: calls provided functions after each iteration step
progressbar: bool: whether to show progressbar or not

Returns

Approximation

Other Parameters

obj_n_mc: `int`: Number of monte carlo samples used for approximation of objective gradients
tf_n_mc: `int`: Number of monte carlo samples used for approximation of test function gradients
obj_optimizer: function (grads, params) -> updates: Optimizer that is used for objective params
test_optimizer: function (grads, params) -> updates: Optimizer that is used for test function params
more_obj_params: `list`: Add custom params for objective optimizer
more_tf_params: `list`: Add custom params for test function optimizer
more_updates: `dict`: Add custom updates to resulting updates
total_grad_norm_constraint: `float`: Bounds gradient norm, prevents exploding gradient problem
fn_kwargs: `dict`: Add kwargs to theano.function (e.g. {‘profile’: True})
more_replacements: `dict`: Apply custom replacements before calculating gradients

refine(n, progressbar=True)¶: Refine the solution using the last compiled step function

class pymc3.variational.inference.KLqp(approx, beta=1.0)¶

Kullback Leibler Divergence Inference

General approach to fit Approximations that define \(logq\) by maximizing ELBO (Evidence Lower Bound). In some cases rescaling the regularization term KL may be beneficial

\[ELBO_\beta = \log p(D|\theta) - \beta KL(q||p)\]

Parameters

approx: :class:`Approximation`: Approximation to fit, it is required to have logQ
beta: float: Scales the regularization term in ELBO (see Christopher P. Burgess et al., 2017)

References

Christopher P. Burgess et al. (NIPS, 2017) Understanding disentangling in \(\beta\)-VAE arXiv preprint 1804.03599

class pymc3.variational.inference.NFVI(*args, **kwargs)¶

Normalizing Flow based :class:`KLqp` inference

Normalizing flow is a series of invertible transformations on initial distribution.

\[z_K = f_K \circ \dots \circ f_2 \circ f_1(z_0)\]

In that case we can compute tractable density for the flow.

\[\ln q_K(z_K) = \ln q_0(z_0) - \sum_{k=1}^{K}\ln \left|\frac{\partial f_k}{\partial z_{k-1}}\right|\]

Every \(f_k\) here is a parametric function with defined determinant. We can choose every step here. For example the here is a simple flow is an affine transform:

\[z = loc(scale(z_0)) = \mu + \sigma * z_0\]

Here we get mean field approximation if \(z_0 \sim \mathcal{N}(0, 1)\)

Flow Formulas

In PyMC3 there is a flexible way to define flows with formulas. We have 5 of them by the moment:

Loc (loc): \(z' = z + \mu\)
Scale (scale): \(z' = \sigma * z\)
Planar (planar): \(z' = z + u * \tanh(w^T z + b)\)
Radial (radial): \(z' = z + \beta (\alpha + (z-z_r))^{-1}(z-z_r)\)
Householder (hh): \(z' = H z\)

Formula can be written as a string, e.g. ‘scale-loc’, ‘scale-hh*4-loc’, ‘panar*10’. Every step is separated with ‘-‘, repeated flow is marked with ‘*’ producing ‘flow*repeats’.

Parameters

flow: str|AbstractFlow: formula or initialized Flow, default is ‘scale-loc’ that is identical to MeanField
model: :class:`pymc3.Model`: PyMC3 model for inference
random_seed: None or int: leave None to use package global RandomStream or other valid value to create instance specific one

class pymc3.variational.inference.SVGD(n_particles=100, jitter=1, model=None, start=None, random_seed=None, estimator=<class 'pymc3.variational.operators.KSD'>, kernel=<pymc3.variational.test_functions.RBF object>, **kwargs)¶

Stein Variational Gradient Descent

This inference is based on Kernelized Stein Discrepancy it’s main idea is to move initial noisy particles so that they fit target distribution best.

Algorithm is outlined below

Input: A target distribution with density function \(p(x)\): and a set of initial particles \(\{x^0_i\}^n_{i=1}\)

Output: A set of particles \(\{x^{*}_i\}^n_{i=1}\) that approximates the target distribution.

\[\begin{split}x_i^{l+1} &\leftarrow x_i^{l} + \epsilon_l \hat{\phi}^{*}(x_i^l) \\ \hat{\phi}^{*}(x) &= \frac{1}{n}\sum^{n}_{j=1}[k(x^l_j,x) \nabla_{x^l_j} logp(x^l_j)+ \nabla_{x^l_j} k(x^l_j,x)]\end{split}\]

Parameters

n_particles: `int`: number of particles to use for approximation
jitter: `float`: noise sd for initial point
model: :class:`pymc3.Model`: PyMC3 model for inference
kernel: `callable`: kernel function for KSD \(f(histogram) -> (k(x,.), \nabla_x k(x,.))\)
temperature: float: parameter responsible for exploration, higher temperature gives more broad posterior estimate
start: `dict`: initial point for inference
random_seed: None or int: leave None to use package global RandomStream or other valid value to create instance specific one
start: `Point`: starting point for inference
kwargs: other keyword arguments passed to estimator

References

Qiang Liu, Dilin Wang (2016) Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm arXiv:1608.04471
Yang Liu, Prajit Ramachandran, Qiang Liu, Jian Peng (2017) Stein Variational Policy Gradient arXiv:1704.02399

pymc3.variational.inference.fit(n=10000, local_rv=None, method='advi', model=None, random_seed=None, start=None, inf_kwargs=None, **kwargs)¶

Handy shortcut for using inference methods in functional way

Parameters

n: `int`

number of iterations

local_rv: dict[var->tuple]

mapping {model_variable -> approx params} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details

method: str or :class:`Inference`

string name is case insensitive in:

‘advi’ for ADVI
‘fullrank_advi’ for FullRankADVI
‘svgd’ for Stein Variational Gradient Descent
‘asvgd’ for Amortized Stein Variational Gradient Descent
‘nfvi’ for Normalizing Flow with default scale-loc flow
‘nfvi=<formula>’ for Normalizing Flow using formula

model: :class:`Model`

PyMC3 model for inference

random_seed: None or int

leave None to use package global RandomStream or other valid value to create instance specific one

inf_kwargs: dict

additional kwargs passed to Inference

start: `Point`

starting point for inference

Returns

Approximation

Other Parameters

score: bool: evaluate loss on each iteration or not
callbacks: list[function: (Approximation, losses, i) -> None]: calls provided functions after each iteration step
progressbar: bool: whether to show progressbar or not
obj_n_mc: `int`: Number of monte carlo samples used for approximation of objective gradients
tf_n_mc: `int`: Number of monte carlo samples used for approximation of test function gradients
obj_optimizer: function (grads, params) -> updates: Optimizer that is used for objective params
test_optimizer: function (grads, params) -> updates: Optimizer that is used for test function params
more_obj_params: `list`: Add custom params for objective optimizer
more_tf_params: `list`: Add custom params for test function optimizer
more_updates: `dict`: Add custom updates to resulting updates
total_grad_norm_constraint: `float`: Bounds gradient norm, prevents exploding gradient problem
fn_kwargs: `dict`: Add kwargs to theano.function (e.g. {‘profile’: True})
more_replacements: `dict`: Apply custom replacements before calculating gradients

Approximations¶

class pymc3.variational.approximations.Empirical(trace=None, size=None, **kwargs)¶

Single Group Full Rank Approximation

Builds Approximation instance from a given trace, it has the same interface as variational approximation

evaluate_over_trace(node)¶

This allows to statically evaluate any symbolic expression over the trace.

Parameters

node: Theano Variables (or Theano expressions)

Returns

evaluated node(s) over the posterior trace contained in the empirical approximation

class pymc3.variational.approximations.FullRank(*args, **kwargs)¶

Single Group Full Rank Approximation

Full Rank approximation to the posterior where Multivariate Gaussian family is fitted to minimize KL divergence from True posterior. In contrast to MeanField approach correlations between variables are taken in account. The main drawback of the method is computational cost.

class pymc3.variational.approximations.MeanField(*args, **kwargs)¶

Single Group Mean Field Approximation

Mean Field approximation to the posterior where spherical Gaussian family is fitted to minimize KL divergence from True posterior. It is assumed that latent space variables are uncorrelated that is the main drawback of the method

class pymc3.variational.approximations.NormalizingFlow(flow='scale-loc', *args, **kwargs)¶

Single Group Normalizing Flow Approximation

Normalizing flow is a series of invertible transformations on initial distribution.

\[\begin{split}z_K &= f_K \circ \dots \circ f_2 \circ f_1(z_0) \\ & z_0 \sim \mathcal{N}(0, 1)\end{split}\]

In that case we can compute tractable density for the flow.

\[\ln q_K(z_K) = \ln q_0(z_0) - \sum_{k=1}^{K}\ln \left|\frac{\partial f_k}{\partial z_{k-1}}\right|\]

Every \(f_k\) here is a parametric function with defined determinant. We can choose every step here. For example the here is a simple flow is an affine transform:

\[z = loc(scale(z_0)) = \mu + \sigma * z_0\]

Here we get mean field approximation if \(z_0 \sim \mathcal{N}(0, 1)\)

Flow Formulas

In PyMC3 there is a flexible way to define flows with formulas. We have 5 of them by the moment:

Loc (loc): \(z' = z + \mu\)
Scale (scale): \(z' = \sigma * z\)
Planar (planar): \(z' = z + u * \tanh(w^T z + b)\)
Radial (radial): \(z' = z + \beta (\alpha + (z-z_r))^{-1}(z-z_r)\)
Householder (hh): \(z' = H z\)

References

Danilo Jimenez Rezende, Shakir Mohamed, 2015 Variational Inference with Normalizing Flows arXiv:1505.05770
Jakub M. Tomczak, Max Welling, 2016 Improving Variational Auto-Encoders using Householder Flow arXiv:1611.09630

pymc3.variational.approximations.sample_approx(approx, draws=100, include_transformed=True)¶

Draw samples from variational posterior.

Parameters

approx: :class:`Approximation`: Approximation to sample from
draws: `int`: Number of random samples.
include_transformed: `bool`: If True, transformed variables are also sampled. Default is True.

Returns

trace: class:pymc3.backends.base.MultiTrace: Samples drawn from variational posterior.

Operators¶

class pymc3.variational.operators.KL(approx, beta=1.0)¶

Operator based on Kullback Leibler Divergence

This operator constructs Evidence Lower Bound (ELBO) objective

\[ELBO_\beta = \log p(D|\theta) - \beta KL(q||p)\]

where

\[KL[q(v)||p(v)] = \int q(v)\log\frac{q(v)}{p(v)}dv\]

Parameters

approx: :class:`Approximation`: Approximation used for inference
beta: float: Beta parameter for KL divergence, scales the regularization term.

apply(f)¶

Operator itself

\[(O^{p,q}f_{\theta})(z)\]

Parameters

f: :class:`TestFunction` or None: function that takes z = self.input and returns same dimensional output

Returns

TensorVariable: symbolically applied operator

class pymc3.variational.operators.KSD(approx, temperature=1)¶

Operator based on Kernelized Stein Discrepancy

Input: A target distribution with density function \(p(x)\): and a set of initial particles \(\{x^0_i\}^n_{i=1}\)

Output: A set of particles \(\{x_i\}^n_{i=1}\) that approximates the target distribution.

\[\begin{split}x_i^{l+1} \leftarrow \epsilon_l \hat{\phi}^{*}(x_i^l) \\ \hat{\phi}^{*}(x) = \frac{1}{n}\sum^{n}_{j=1}[k(x^l_j,x) \nabla_{x^l_j} logp(x^l_j)/temp + \nabla_{x^l_j} k(x^l_j,x)]\end{split}\]

Parameters

approx: :class:`Approximation`: Approximation used for inference
temperature: float: Temperature for Stein gradient

References

Qiang Liu, Dilin Wang (2016) Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm arXiv:1608.04471

apply(f)¶

Operator itself

\[(O^{p,q}f_{\theta})(z)\]

Parameters

f: :class:`TestFunction` or None: function that takes z = self.input and returns same dimensional output

Returns

TensorVariable: symbolically applied operator

objective_class¶: alias of pymc3.variational.operators.KSDObjective