# Inference¶

## Sampling¶

pymc3.sampling.sample(draws=500, step=None, init='auto', n_init=200000, start=None, trace=None, chain=0, njobs=1, tune=500, nuts_kwargs=None, step_kwargs=None, progressbar=True, model=None, random_seed=-1, live_plot=False, discard_tuned_samples=True, **kwargs)

Draw samples from the posterior using the given step methods.

Multiple step methods are supported via compound step methods.

Examples

>>> import pymc3 as pm
... n = 100
... h = 61
... alpha = 2
... beta = 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(2000, tune=1000, njobs=4)
>>> pm.df_summary(trace)
mean        sd  mc_error   hpd_2.5  hpd_97.5
p  0.604625  0.047086   0.00078  0.510498  0.694774
pymc3.sampling.iter_sample(draws, step, start=None, trace=None, chain=0, tune=None, model=None, random_seed=-1)

Generator that returns a trace on each iteration using the given step method. Multiple step methods supported via compound step method returns the amount of time taken.

Parameters: draws (int) – The number of samples to draw step (function) – Step function start (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) trace (backend, 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 (int) – Chain number used to store sample in backend. If njobs is greater than one, chain numbers will start here. tune (int) – Number of iterations to tune, if applicable (defaults to None) model (Model (optional if in with context)) – random_seed (int or list of ints) – A list is accepted if more if njobs is greater than one.

Example

for trace in iter_sample(500, step):
...
pymc3.sampling.sample_ppc(trace, samples=None, model=None, vars=None, size=None, random_seed=None, progressbar=True)

Generate posterior predictive samples from a model given a trace.

Parameters: trace (backend, list, or MultiTrace) – Trace generated from MCMC sampling samples (int) – Number of posterior predictive samples to generate. Defaults to the length of trace model (Model (optional if in with context)) – Model used to generate trace vars (iterable) – Variables for which to compute the posterior predictive samples. Defaults to model.observed_RVs. size (int) – The number of random draws from the distribution specified by the parameters in each sample of the trace. samples (dict) – Dictionary with the variables as keys. The values corresponding to the posterior predictive samples.
pymc3.sampling.init_nuts(init='ADVI', njobs=1, n_init=500000, model=None, random_seed=-1, progressbar=True, **kwargs)

Initialize and sample from posterior of a continuous model.

This is a convenience function. NUTS convergence and sampling speed is extremely dependent on the choice of mass/scaling matrix. In our experience, using ADVI to estimate a diagonal covariance matrix and using this as the scaling matrix produces robust results over a wide class of continuous models.

Parameters: init (str {'ADVI', 'ADVI_MAP', 'MAP', 'NUTS'}) – Initialization method to use. * ADVI : Run ADVI to estimate posterior mean and diagonal covariance matrix. * ADVI_MAP: Initialize ADVI with MAP and use MAP as starting point. * MAP : Use the MAP as starting point. * NUTS : Run NUTS and estimate posterior mean and covariance matrix. njobs (int) – Number of parallel jobs to start. n_init (int) – Number of iterations of initializer If ‘ADVI’, number of iterations, if ‘metropolis’, number of draws. model (Model (optional if in with context)) – progressbar (bool) – Whether or not to display a progressbar for advi sampling. **kwargs (keyword arguments) – Extra keyword arguments are forwarded to pymc3.NUTS. start (pymc3.model.Point) – Starting point for sampler nuts_sampler (pymc3.step_methods.NUTS) – Instantiated and initialized NUTS sampler object

## Step-methods¶

### NUTS¶

class pymc3.step_methods.hmc.nuts.NUTS(vars=None, Emax=1000, target_accept=0.8, gamma=0.05, k=0.75, t0=10, adapt_step_size=True, max_treedepth=10, **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.

References

 [1] Hoffman, Matthew D., & Gelman, Andrew. (2011). The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo.
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 (0,1), default .8) – Try to find a 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. 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 (5,1) default .75) – scaling of speed of adaptation t0 (int, default 10) – slows 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 stoped when this depth is reached. integrator (str, default "leapfrog") – The integrator to use for the trajectories. One of “leapfrog”, “two-stage” or “three-stage”. The second two can increase sampling speed for some high dimensional problems. 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.

check_trace(strace)

Print warnings for obviously problematic chains.

### Metropolis¶

class pymc3.step_methods.metropolis.Metropolis(vars=None, S=None, proposal_dist=None, scaling=1.0, tune=True, tune_interval=100, model=None, mode=None, **kwargs)

Metropolis-Hastings sampling step

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
class pymc3.step_methods.metropolis.BinaryMetropolis(vars, scaling=1.0, tune=True, tune_interval=100, model=None)

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.BinaryGibbsMetropolis(vars, order='random', model=None)

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

static competence(var)

BinaryMetropolis is only suitable for Bernoulli and Categorical variables with k=2.

class pymc3.step_methods.metropolis.CategoricalGibbsMetropolis(vars, proposal='uniform', order='random', model=None)

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.

### Slice¶

class pymc3.step_methods.slicer.Slice(vars=None, w=1.0, tune=True, model=None, **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(vars=None, path_length=2.0, step_rand=<function unif>, **kwargs)
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. model (pymc3.Model) – The model **kwargs (passed to BaseHMC) –

## Variational¶

### 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 inverence 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 forth 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

class pymc3.variational.opvi.ObjectiveFunction(op, tf)

Helper class for construction loss and updates for variational inference

Parameters: op (Operator) – OPVI Functional operator tf (TestFunction) – OPVI TestFunction
random(size=None)

Posterior distribution from initial latent space

Parameters: size (int) – number of samples from distribution posterior space (theano)
score_function(sc_n_mc=None, more_replacements=None, fn_kwargs=None)

Compiles scoring function that operates which takes no inputs and returns Loss

Parameters: sc_n_mc (int) – number of scoring MC samples more_replacements – Apply custom replacements before compiling a function fn_kwargs (dict) – arbitrary kwargs passed to theano.function theano.function

Step function that should be called on each optimization step.

Generally it solves the following problem:

$\mathbf{\lambda^{*}} = \inf_{\lambda} \sup_{\theta} t(\mathbb{E}_{\lambda}[(O^{p,q}f_{\theta})(z)])$

Calculates gradients for objective function, test function and then constructs updates for optimization step

class pymc3.variational.opvi.Operator(approx)

Base class for Operator

Parameters: approx (Approximation) – an approximation instance

Notes

For implementing Custom operator it is needed to define Operator.apply() method

OBJECTIVE

alias of ObjectiveFunction

apply(f)

Operator itself

$(O^{p,q}f_{\theta})(z)$
Parameters: f (TestFunction or None) – function that takes z = self.input and returns same dimensional output TensorVariable – symbolically applied operator
class pymc3.variational.opvi.Approximation(local_rv=None, model=None, cost_part_grad_scale=1, scale_cost_to_minibatch=False, random_seed=None, **kwargs)

Base class for approximations.

Parameters: local_rv (dict[var->tuple]) – mapping {model_variable -> local_variable ($$\mu$$, $$\rho$$)} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details model (Model) – PyMC3 model for inference cost_part_grad_scale (float or scalar tensor) – Scaling score part of gradient can be useful near optimum for archiving better convergence properties. Common schedule is 1 at the start and 0 in the end. So slow decay will be ok. See (Sticking the Landing; Geoffrey Roeder, Yuhuai Wu, David Duvenaud, 2016) for details scale_cost_to_minibatch (bool, default False) – Scale cost to minibatch instead of full dataset random_seed (None or int) – leave None to use package global RandomStream or other valid value to create instance specific one

Notes

Defining an approximation needs custom implementation of the following methods:

• .create_shared_params(**kwargs)
Returns {dict|list|theano.shared}
• .random_global(size=None, no_rand=False)
Generate samples from posterior. If no_rand==False: sample from MAP of initial distribution. Returns TensorVariable
• .log_q_W_global(z)
It is needed only if used with operator that requires $$logq$$ of an approximation Returns Scalar

You can also override the following methods:

• ._setup(**kwargs) Do some specific stuff having kwargs before calling Approximation.create_shared_params()
• .check_model(model, **kwargs) Do some specific check for model having kwargs

kwargs mentioned above are supplied as additional arguments for Approximation

There are some defaults class attributes for approximation classes that can be optionally overridden.

• initial_dist_name string that represents name of the initial distribution. In most cases if will be uniform or normal
• initial_dist_map float where initial distribution has maximum density

References

• 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.
apply_replacements(node, deterministic=False, include=None, exclude=None, more_replacements=None)

Replace variables in graph with variational approximation. By default, replaces all variables

Parameters: node (Theano Variables (or Theano expressions)) – node or nodes for replacements deterministic (bool) – whether to use zeros as initial distribution if True - zero initial point will produce constant latent variables include (list) – latent variables to be replaced exclude (list) – latent variables to be excluded for replacements more_replacements (dict) – add custom replacements to graph, e.g. change input source node(s) with replacements
check_model(model, **kwargs)

Checks that model is valid for variational inference

construct_replacements(include=None, exclude=None, more_replacements=None)

Construct replacements with given conditions

Parameters: include (list) – latent variables to be replaced exclude (list) – latent variables to be excluded for replacements more_replacements (dict) – add custom replacements to graph, e.g. change input source dict – Replacements
create_shared_params(**kwargs)
Returns: {dict|list|theano.shared}
initial(size, no_rand=False, l=None)

Initial distribution for constructing posterior

Parameters: size (int) – number of samples no_rand (bool) – return zeros if True l (int) – length of sample, defaults to latent space dim tt.TensorVariable – sampled latent space
log_q_W_global(z)

log_q_W samples over q for global vars

log_q_W_local(z)

log_q_W samples over q for local vars Gradient wrt mu, rho in density parametrization can be scaled to lower variance of ELBO

logq(z)

Total logq for approximation

random(size=None, no_rand=False)

Implements posterior distribution from initial latent space

Parameters: size (scalar) – number of samples from distribution no_rand (bool) – whether use deterministic distribution posterior space (theano)
random_fn

Implements posterior distribution from initial latent space

Parameters: size (int) – number of samples from distribution no_rand (bool) – whether use deterministic distribution posterior space (numpy)
random_global(size=None, no_rand=False)

Implements posterior distribution from initial latent space

Parameters: size (scalar) – number of samples from distribution no_rand (bool) – whether use deterministic distribution global posterior space
random_local(size=None, no_rand=False)

Implements posterior distribution from initial latent space

Parameters: size (scalar) – number of samples from distribution no_rand (bool) – whether use deterministic distribution local posterior space
sample(draws=1, include_transformed=False)

Draw samples from variational posterior.

Parameters: draws (int) – Number of random samples. include_transformed (bool) – If True, transformed variables are also sampled. Default is False. trace (pymc3.backends.base.MultiTrace) – Samples drawn from variational posterior.
sample_node(node, size=100, more_replacements=None)

Samples given node or nodes over shared posterior

Parameters: node (Theano Variables (or Theano expressions)) – size (scalar) – number of samples more_replacements (dict) – add custom replacements to graph, e.g. change input source sampled node(s) with replacements

References

• Geoffrey Roeder, Yuhuai Wu, David Duvenaud, 2016
seed(random_seed=None)

Reinitialize RandomStream used by this approximation

Parameters: random_seed (int) – New random seed
to_flat_input(node)

Replaces vars with flattened view stored in self.input

view(space, name, reshape=True)

Construct view on a variable from flattened space

Parameters: space (matrix or vector) – space to take view of variable from name (str) – name of variable reshape (bool) – whether to reshape variable from vectorized view (reshaped) slice of matrix – variable view

### Inference¶

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.

Parameters: local_rv (dict[var->tuple]) – mapping {model_variable -> local_variable ($$\mu$$, $$\rho$$)} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details model (pymc3.Model) – PyMC3 model for inference cost_part_grad_scale (scalar) – Scaling score part of gradient can be useful near optimum for archiving better convergence properties. Common schedule is 1 at the start and 0 in the end. So slow decay will be ok. See (Sticking the Landing; Geoffrey Roeder, Yuhuai Wu, David Duvenaud, 2016) for details scale_cost_to_minibatch (bool) – Scale cost to minibatch instead of full dataset, default False 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.
classmethod from_mean_field(mean_field)

Full Rank Automatic Differentiation Variational Inference (ADVI)

Parameters: local_rv (dict[var->tuple]) – mapping {model_variable -> local_variable ($$\mu$$, $$\rho$$)} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details model (pymc3.Model) – PyMC3 model for inference cost_part_grad_scale (scalar) – Scaling score part of gradient can be useful near optimum for archiving better convergence properties. Common schedule is 1 at the start and 0 in the end. So slow decay will be ok. See (Sticking the Landing; Geoffrey Roeder, Yuhuai Wu, David Duvenaud, 2016) for details scale_cost_to_minibatch (bool, default False) – Scale cost to minibatch instead of full dataset 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.

classmethod from_full_rank(full_rank)

classmethod from_mean_field(mean_field, gpu_compat=False)

Parameters: Other Parameters: mean_field (MeanField) – approximation to start with gpu_compat (bool) – use GPU compatible version or not FullRankADVI
class pymc3.variational.inference.SVGD(n_particles=100, jitter=0.01, model=None, kernel=<pymc3.variational.test_functions.RBF object>, scale_cost_to_minibatch=False, start=None, histogram=None, random_seed=None, local_rv=None)

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 (pymc3.Model) – PyMC3 model for inference kernel (callable) – kernel function for KSD $$f(histogram) -> (k(x,.), \nabla_x k(x,.))$$ scale_cost_to_minibatch (bool, default False) – Scale cost to minibatch instead of full dataset start (Point) – initial point for inference histogram (Empirical) – initialize SVGD with given Empirical approximation instead of default initial particles random_seed (None or int) – leave None to use package global RandomStream or other valid value to create instance specific one start – starting point for inference

References

• Qiang Liu, Dilin Wang (2016) Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm arXiv:1608.04471
class pymc3.variational.inference.ASVGD(approx=<class 'pymc3.variational.approximations.FullRank'>, local_rv=None, kernel=<pymc3.variational.test_functions.RBF object>, model=None, **kwargs)

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 (Approximation) – local_rv (dict[var->tuple]) – mapping {model_variable -> local_variable ($$\mu$$, $$\rho$$)} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details kernel (callable) – kernel function for KSD $$f(histogram) -> (k(x,.), \nabla_x k(x,.))$$ model (Model) – kwargs (kwargs for Approximation) –

References

fit(n=10000, score=None, callbacks=None, progressbar=True, obj_n_mc=30, **kwargs)

Performs Amortized Stein Variational Gradient Descent

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 obj_n_mc (int) – sample n particles for Stein gradient kwargs (kwargs) – additional kwargs for ObjectiveFunction.step_function() Approximation
class pymc3.variational.inference.Inference(op, approx, tf, local_rv=None, model=None, **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) – local_rv (dict) – mapping {model_variable -> local_variable} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details model (Model) – PyMC3 Model kwargs (kwargs) – additional kwargs for Approximation
fit(n=10000, score=None, callbacks=None, progressbar=True, **kwargs)

Performs 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 kwargs (kwargs) – additional kwargs for ObjectiveFunction.step_function() Approximation
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: Other Parameters: n (int) – number of iterations local_rv (dict[var->tuple]) – mapping {model_variable -> local_variable ($$\mu$$, $$\rho$$)} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details method (str or Inference) – string name is case insensitive in {‘advi’, ‘fullrank_advi’, ‘advi->fullrank_advi’, ‘svgd’, ‘asvgd’} model (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 inf_kwargs (dict) – additional kwargs passed to Inference start (Point) – starting point for inference frac (float) – if method is ‘advi->fullrank_advi’ represents advi fraction when training kwargs (kwargs) – additional kwargs for Inference.fit() Approximation

### Approximations¶

class pymc3.variational.approximations.MeanField(local_rv=None, model=None, cost_part_grad_scale=1, scale_cost_to_minibatch=False, random_seed=None, **kwargs)

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

Parameters: local_rv (dict[var->tuple]) – mapping {model_variable -> local_variable ($$\mu$$, $$\rho$$)} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details model (pymc3.Model) – PyMC3 model for inference start (Point) – initial mean cost_part_grad_scale (scalar) – Scaling score part of gradient can be useful near optimum for archiving better convergence properties. Common schedule is 1 at the start and 0 in the end. So slow decay will be ok. See (Sticking the Landing; Geoffrey Roeder, Yuhuai Wu, David Duvenaud, 2016) for details scale_cost_to_minibatch (bool) – Scale cost to minibatch instead of full dataset, default False random_seed (None or int) – leave None to use package global RandomStream or other valid value to create instance specific one

References

• Geoffrey Roeder, Yuhuai Wu, David Duvenaud, 2016 Sticking the Landing: A Simple Reduced-Variance Gradient for ADVI approximateinference.org/accepted/RoederEtAl2016.pdf
log_q_W_global(z)

log_q_W samples over q for global vars

class pymc3.variational.approximations.FullRank(local_rv=None, model=None, cost_part_grad_scale=1, scale_cost_to_minibatch=False, gpu_compat=False, random_seed=None, **kwargs)

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.

Parameters: Other Parameters: local_rv (dict[var->tuple]) – mapping {model_variable -> local_variable ($$\mu$$, $$\rho$$)} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details model (PyMC3 model for inference) – start (Point) – initial mean cost_part_grad_scale (float or scalar tensor) – Scaling score part of gradient can be useful near optimum for archiving better convergence properties. Common schedule is 1 at the start and 0 in the end. So slow decay will be ok. See (Sticking the Landing; Geoffrey Roeder, Yuhuai Wu, David Duvenaud, 2016) for details scale_cost_to_minibatch (bool, default False) – Scale cost to minibatch instead of full dataset random_seed (None or int) – leave None to use package global RandomStream or other valid value to create instance specific one gpu_compat (bool) – use GPU compatible version or not

References

• Geoffrey Roeder, Yuhuai Wu, David Duvenaud, 2016 Sticking the Landing: A Simple Reduced-Variance Gradient for ADVI approximateinference.org/accepted/RoederEtAl2016.pdf
classmethod from_mean_field(mean_field, gpu_compat=False)

Construct FullRank from MeanField approximation

Parameters: Other Parameters: mean_field (MeanField) – approximation to start with gpu_compat (bool) – use GPU compatible version or not FullRank
log_q_W_global(z)

log_q_W samples over q for global vars

class pymc3.variational.approximations.Empirical(trace, local_rv=None, scale_cost_to_minibatch=False, model=None, random_seed=None, **kwargs)

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

Parameters: trace (MultiTrace) – Trace storing samples (e.g. from step methods) local_rv (dict[var->tuple]) – Experimental for Empirical Approximation mapping {model_variable -> local_variable ($$\mu$$, $$\rho$$)} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details scale_cost_to_minibatch (bool) – Scale cost to minibatch instead of full dataset, default False model (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

Examples

>>> with model:
...     step = NUTS()
...     trace = sample(1000, step=step)
...     histogram = Empirical(trace[100:])
classmethod from_noise(size, jitter=0.01, local_rv=None, start=None, model=None, random_seed=None, **kwargs)

Initialize Histogram with random noise

Parameters: size (int) – number of initial particles jitter (float) – initial sd local_rv (dict) – mapping {model_variable -> local_variable} Local Vars are used for Autoencoding Variational Bayes See (AEVB; Kingma and Welling, 2014) for details start (Point) – initial point model (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 kwargs (other kwargs passed to init) – Empirical
histogram

Shortcut to flattened Trace

histogram_logp

Symbolic logp for every point in trace

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

Draw samples from variational posterior.

Parameters: approx (Approximation) – Approximation to sample from draws (int) – Number of random samples. include_transformed (bool) – If True, transformed variables are also sampled. Default is True. trace (class:pymc3.backends.base.MultiTrace) – Samples drawn from variational posterior.

### Operators¶

class pymc3.variational.operators.KL(approx)

Operator based on Kullback Leibler Divergence

$KL[q(v)||p(v)] = \int q(v)\log\frac{q(v)}{p(v)}dv$
class pymc3.variational.operators.KSD(approx)

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.

$\begin{split}x_i^{l+1} \leftarrow \epsilon_l \hat{\phi}^{*}(x_i^l) \\ \hat{\phi}^{*}(x) = \frac{1}{n}\sum^{n}_{j=1}[k(x^l_j,x) \nabla_{x^l_j} logp(x^l_j)+ \nabla_{x^l_j} k(x^l_j,x)]\end{split}$
Parameters: approx (Empirical) – Empirical Approximation used for inference

References

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

alias of KSDObjective