# PyMC Developer Guide#

PyMC is a Python package for Bayesian statistical modeling built on top of Aesara. This document aims to explain the design and implementation of probabilistic programming in PyMC, with comparisons to other PPLs like TensorFlow Probability (TFP) and Pyro. A user-facing API introduction can be found in the API quickstart. A more accessible, user facing deep introduction can be found in Peadar Coyle’s probabilistic programming primer.

## Distribution#

Probability distributions in PyMC are implemented as classes that inherit from Continuous or Discrete. Either of these inherit Distribution which defines the high level API.

For a detailed introduction on how a new distribution should be implemented check out the guide on implementing distributions.

## Reflection#

How tensor/value semantics for probability distributions are enabled in PyMC:

In PyMC, model variables are defined by calling probability distribution classes with parameters:

z = Normal("z", 0, 5)


This is done inside the context of a pm.Model, which intercepts some information, for example to capture known dimensions. The notation aligns with the typically used math notation:

$z \sim \text{Normal}(0, 5)$

A call to a Distribution constructor as shown above returns an Aesara TensorVariable, which is a symbolic representation of the model variable and the graph of inputs it depends on. Under the hood, the variables are created through the dist() API, which calls the RandomVariable Op corresponding to the distribution.

At a high level of abstraction, the idea behind RandomVariable Ops is to create symbolic variables (TensorVariables) that can be associated with the properties of a probability distribution. For example, the RandomVariable Op which becomes part of the symbolic computation graph is associated with the random number generators or probability mass/density functions of the distribution.

In the example above, where the TensorVariable z is created to be $$\text{Normal}(0, 5)$$ random variable, we can get a handle on the corresponding RandomVariable Op instance:

with pm.Model():
z = pm.Normal("z", 0, 5)
print(type(z.owner.op))
# ==> aesara.tensor.random.basic.NormalRV
isinstance(z.owner.op, aesara.tensor.random.basic.RandomVariable)
# ==> True


Now, because the NormalRV can be associated with the probability density function of the Normal distribution, we can now evaluate it through the special pm.logp function:

with pm.Model():
z = pm.Normal("z", 0, 5)
symbolic = pm.logp(z, 2.5)
numeric = symbolic.eval()
# array(-2.65337645)


We can, of course, also work out the math by hand:

\begin{split} \begin{aligned} pdf_{\mathcal{N}}(\mu, \sigma, x) &= \frac{1}{\sigma \sqrt{2 \pi}} \exp^{- 0.5 (\frac{x - \mu}{\sigma})^2} \\ pdf_{\mathcal{N}}(0, 5, 0.3) &= 0.070413 \\ ln(0.070413) &= -2.6533 \end{aligned} \end{split}

In the probabilistic programming context, this enables PyMC and its backend libraries aeppl and Aesara to create and evaluate computation graphs to compute, for example log-prior or log-likelihood values.

## PyMC in Comparison#

Within the PyMC model context, random variables are essentially Aesara tensors that can be used in all kinds of operations as if they were NumPy arrays. This is different compared to TFP and pyro, where one needs to be more explicit about the conversion from random variables to tensors.

Consider the following examples, which implement the below model.

\begin{split} \begin{aligned} z &\sim \mathcal{N}(0, 5) \\ x &\sim \mathcal{N}(z, 1) \\ \end{aligned} \end{split}

### PyMC#

with pm.Model() as model:
z = pm.Normal('z', mu=0., sigma=5.)             # ==> aesara.tensor.var.TensorVariable
x = pm.Normal('x', mu=z, sigma=1., observed=5.) # ==> aesara.tensor.var.TensorVariable
# The log-prior of z=2.5
pm.logp(z, 2.5).eval()                              # ==> -2.65337645
# ???????
x.logp({'z': 2.5})                                  # ==> -4.0439386
# ???????
model.logp({'z': 2.5})                              # ==> -6.6973152


### Tensorflow Probability#


import tensorflow.compat.v1 as tf
from tensorflow_probability import distributions as tfd

with tf.Session() as sess:
z_dist = tfd.Normal(loc=0., scale=5.)            # ==> <class 'tfp.python.distributions.normal.Normal'>
z = z_dist.sample()                              # ==> <class 'tensorflow.python.framework.ops.Tensor'>
x = tfd.Normal(loc=z, scale=1.).log_prob(5.)     # ==> <class 'tensorflow.python.framework.ops.Tensor'>
model_logp = z_dist.log_prob(z) + x
print(sess.run(x, feed_dict={z: 2.5}))           # ==> -4.0439386
print(sess.run(model_logp, feed_dict={z: 2.5}))  # ==> -6.6973152


### Pyro#

z_dist = dist.Normal(loc=0., scale=5.)           # ==> <class 'pyro.distributions.torch.Normal'>
z = pyro.sample("z", z_dist)                     # ==> <class 'torch.Tensor'>
# reset/specify value of z
z.data = torch.tensor(2.5)
x = dist.Normal(loc=z, scale=1.).log_prob(5.)    # ==> <class 'torch.Tensor'>
model_logp = z_dist.log_prob(z) + x
x                                                # ==> -4.0439386
model_logp                                       # ==> -6.6973152


## Behind the scenes of the logp function#

The logp function is straightforward - it is an Aesara function within each distribution. It has the following signature:

Warning

The code block is outdated.

def logp(self, value):
# GET PARAMETERS
param1, param2, ... = self.params1, self.params2, ...
# EVALUATE LOG-LIKELIHOOD FUNCTION, all inputs are (or array that could be convert to) Aesara tensor
total_log_prob = f(param1, param2, ..., value)


In the logp method, parameters and values are either Aesara tensors, or could be converted to tensors. It is rather convenient as the evaluation of logp is represented as a tensor (RV.logpt), and when we linked different logp together (e.g., summing all RVs.logpt to get the model total logp) the dependence is taken care of by Aesara when the graph is built and compiled. Again, since the compiled function depends on the nodes that already in the graph, whenever you want to generate a new function that takes new input tensors you either need to regenerate the graph with the appropriate dependencies, or replace the node by editing the existing graph. In PyMC we use the second approach by using aesara.clone_replace() when it is needed.

As explained above, distribution in a pm.Model() context automatically turn into a tensor with distribution property (PyMC random variable). To get the logp of a free_RV is just evaluating the logp() on itself:

# self is a aesara.tensor with a distribution attached
self.logp_sum_unscaledt = distribution.logp_sum(self)
self.logp_nojac_unscaledt = distribution.logp_nojac(self)


Or for an observed RV. it evaluate the logp on the data:

self.logp_sum_unscaledt = distribution.logp_sum(data)
self.logp_nojac_unscaledt = distribution.logp_nojac(data)


### Model context and Random Variable#

I like to think that the with pm.Model() ... is a key syntax feature and the signature of PyMC model language, and in general a great out-of-the-box thinking/usage of the context manager in Python (with some critics, of course).

Essentially what a context manager does is:

with EXPR as VAR:
USERCODE


which roughly translates into this:

VAR = EXPR
VAR.__enter__()
try:
USERCODE
finally:
VAR.__exit__()


or conceptually:

with EXPR as VAR:
# DO SOMETHING
USERCODE


So what happened within the with pm.Model() as model: ... block, besides the initial set up model = pm.Model()? Starting from the most elementary:

### Random Variable#

From the above session, we know that when we call e.g. pm.Normal('x', ...) within a Model context, it returns a random variable. Thus, we have two equivalent ways of adding random variable to a model:

with pm.Model() as m:
x = pm.Normal('x', mu=0., sigma=1.)

print(type(x))                              # ==> <class 'aesara.tensor.var.TensorVariable'>
print(m.free_RVs)                           # ==> [x]
print(logpt(x, 5.0))                        # ==> Elemwise{switch,no_inplace}.0
print(logpt(x, 5.).eval({}))                # ==> -13.418938533204672
print(m.logp({'x': 5.}))                    # ==> -13.418938533204672


In general, if a variable has observations (observed parameter), the RV is an observed RV, otherwise if it has a transformed (transform parameter) attribute, it is a transformed RV otherwise, it will be the most elementary form: a free RV. Note that this means that random variables with observations cannot be transformed.

### Additional things that pm.Model does#

In a way, pm.Model is a tape machine that records what is being added to the model, it keeps track the random variables (observed or unobserved) and potential term (additional tensor that to be added to the model logp), and also deterministic transformation (as bookkeeping):

• named_vars

• free_RVs

• observed_RVs

• deterministics

• potentials

• missing_values The model context then computes some simple model properties, builds a bijection mapping that transforms between dictionary and numpy/Aesara ndarray, thus allowing the logp/dlogp functions to have two equivalent versions: One takes a dict as input and the other takes an ndarray as input. More importantly, a pm.Model() contains methods to compile Aesara functions that take Random Variables (that are also initialised within the same model) as input, for example:

with pm.Model() as m:
z = pm.Normal('z', 0., 10., shape=10)
x = pm.Normal('x', z, 1., shape=10)

print(m.initial_point)
print(m.dict_to_array(m.initial_point))  # ==> m.bijection.map(m.initial_point)
print(m.bijection.rmap(np.arange(20)))
# {'z': array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.]), 'x': array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])}
# [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
# {'z': array([10., 11., 12., 13., 14., 15., 16., 17., 18., 19.]), 'x': array([0., 1., 2., 3., 4., 5., 6., 7., 8., 9.])}

list(filter(lambda x: "logp" in x, dir(pm.Model)))
#['d2logp',
# 'd2logp_nojac',
# 'datalogpt',
# 'dlogp',
# 'dlogp_array',
# 'dlogp_nojac',
# 'fastd2logp',
# 'fastd2logp_nojac',
# 'fastdlogp',
# 'fastdlogp_nojac',
# 'fastlogp',
# 'fastlogp_nojac',
# 'logp',
# 'logp_array',
# 'logp_dlogp_function',
# 'logp_elemwise',
# 'logp_nojac',
# 'logp_nojact',
# 'logpt',
# 'varlogpt']


### Logp and dlogp#

The model collects all the random variables (everything in model.free_RVs and model.observed_RVs) and potential term, and sum them together to get the model logp:

@property
def logpt(self):
"""Aesara scalar of log-probability of the model"""
with self:
factors = [var.logpt for var in self.basic_RVs] + self.potentials
logp = at.sum([at.sum(factor) for factor in factors])
...
return logp


which returns an Aesara tensor that its value depends on the free parameters in the model (i.e., its parent nodes from the Aesara graph). You can evaluate or compile into a python callable (that you can pass numpy as input args). Note that the logp tensor depends on its input in the Aesara graph, thus you cannot pass new tensor to generate a logp function. For similar reason, in PyMC we do graph copying a lot using aesara.clone_replace to replace the inputs to a tensor.

with pm.Model() as m:
z = pm.Normal('z', 0., 10., shape=10)
x = pm.Normal('x', z, 1., shape=10)
y = pm.Normal('y', x.sum(), 1., observed=2.5)

print(m.basic_RVs)    # ==> [z, x, y]
print(m.free_RVs)     # ==> [z, x]

type(m.logpt)
# aesara.tensor.var.TensorVariable

m.logpt.eval({x: np.random.randn(*x.tag.test_value.shape) for x in m.free_RVs})
# array(-51.25369126)


PyMC then compiles a logp function with gradient that takes model.free_RVs as input and model.logpt as output. It could be a subset of tensors in model.free_RVs if we want a conditional logp/dlogp function:

def logp_dlogp_function(self, grad_vars=None, **kwargs):
else:
...
varnames = [var.name for var in grad_vars]  # In a simple case with only continous RVs,
# this is all the free_RVs
extra_vars = [var for var in self.free_RVs if var.name not in varnames]


ValueGradFunction is a callable class which isolates part of the Aesara graph to compile additional Aesara functions. PyMC relies on aesara.clone_replace to copy the model.logpt and replace its input. It does not edit or rewrite the graph directly.

The important parts of the above function is highlighted and commented. On a high level, it allows us to build conditional logp function and its gradient easily. Here is a taste of how it works in action:

inputlist = [np.random.randn(*x.tag.test_value.shape) for x in m.free_RVs]

func = m.logp_dlogp_function()
func.set_extra_values({})
input_dict = {x.name: y for x, y in zip(m.free_RVs, inputlist)}
print(input_dict)
input_array = func.dict_to_array(input_dict)
print(input_array)
print(" ===== ")
func(input_array)

# {'z': array([-0.7202002 ,  0.58712205, -1.44120196, -0.53153001, -0.36028732,
#         -1.49098414, -0.80046792, -0.26351819,  1.91841949,  1.60004128]), 'x': array([ 0.01490006,  0.60958275, -0.06955203, -0.42430833, -1.43392303,
#         1.13713493,  0.31650495, -0.62582879,  0.75642811,  0.50114527])}
# [-0.7202002   0.58712205 -1.44120196 -0.53153001 -0.36028732 -1.49098414
#     -0.80046792 -0.26351819  1.91841949  1.60004128  0.01490006  0.60958275
#     -0.06955203 -0.42430833 -1.43392303  1.13713493  0.31650495 -0.62582879
#     0.75642811  0.50114527]
#     =====
# (array(-51.0769075),
#     array([ 0.74230226,  0.01658948,  1.38606194,  0.11253699, -1.07003284,
#             2.64302891,  1.12497754, -0.35967542, -1.18117557, -1.11489642,
#             0.98281586,  1.69545542,  0.34626619,  1.61069443,  2.79155183,
#         -0.91020295,  0.60094326,  2.08022672,  2.8799075 ,  2.81681213]))

irv = 1
print("Condition Logp: take %s as input and conditioned on the rest."%(m.free_RVs[irv].name))
func_conditional.set_extra_values(input_dict)
input_array2 = func_conditional.dict_to_array(input_dict)
print(input_array2)
print(" ===== ")
func_conditional(input_array2)

# Condition Logp: take x as input and conditioned on the rest.
# [ 0.01490006  0.60958275 -0.06955203 -0.42430833 -1.43392303  1.13713493
#     0.31650495 -0.62582879  0.75642811  0.50114527]
#     =====
# (array(-51.0769075),
#     array([ 0.98281586,  1.69545542,  0.34626619,  1.61069443,  2.79155183,
#         -0.91020295,  0.60094326,  2.08022672,  2.8799075 ,  2.81681213]))


So why is this necessary? One can imagine that we just compile one logp function, and do bookkeeping ourselves. For example, we can build the logp function in Aesara directly:

import aesara
func = aesara.function(m.free_RVs, m.logpt)
func(*inputlist)
# array(-51.0769075)

func_d(*inputlist)
# [array([ 0.74230226,  0.01658948,  1.38606194,  0.11253699, -1.07003284,
#          2.64302891,  1.12497754, -0.35967542, -1.18117557, -1.11489642]),
#  array([ 0.98281586,  1.69545542,  0.34626619,  1.61069443,  2.79155183,
#         -0.91020295,  0.60094326,  2.08022672,  2.8799075 ,  2.81681213])]


Similarly, build a conditional logp:

shared = aesara.shared(inputlist[1])
func2 = aesara.function([m.free_RVs[0]], m.logpt, givens=[(m.free_RVs[1], shared)])
print(func2(inputlist[0]))
# -51.07690750130328

func_d2 = aesara.function([m.free_RVs[0]], logpt_grad2, givens=[(m.free_RVs[1], shared)])
print(func_d2(inputlist[0]))
# [ 0.74230226  0.01658948  1.38606194  0.11253699 -1.07003284  2.64302891
#   1.12497754 -0.35967542 -1.18117557 -1.11489642]


The above also gives the same logp and gradient as the output from model.logp_dlogp_function. But the difficulty is to compile everything into a single function:

func_logp_and_grad = aesara.function(m.free_RVs, [m.logpt, logpt_grad])
# ==> ERROR


We want to have a function that return the evaluation and its gradient re each input: value, grad = f(x), but the naive implementation does not work. We can of course wrap 2 functions - one for logp one for dlogp - and output a list. But that would mean we need to call 2 functions. In addition, when we write code using python logic to do bookkeeping when we build our conditional logp. Using aesara.clone_replace, we always have the input to the Aesara function being a 1d vector (instead of a list of RV that each can have very different shape), thus it is very easy to do matrix operation like rotation etc.

### Notes#

The current setup is quite powerful, as the Aesara compiled function is fairly fast to compile and to call. Also, when we are repeatedly calling a conditional logp function, external RV only need to reset once. However, there are still significant overheads when we are passing values between Aesara graph and NumPy. That is the reason we often see no advantage in using GPU, because the data is copying between GPU and CPU at each function call - and for a small model, the result is a slower inference under GPU than CPU.

Also, aesara.clone_replace is too convenient (PyMC internal joke is that it is like a drug - very addictive). If all the operation happens in the graph (including the conditioning and setting value), I see no need to isolate part of the graph (via graph copying or graph rewriting) for building model and running inference.

Moreover, if we are limiting to the problem that we can solved most confidently - model with all continous unknown parameters that could be sampled with dynamic HMC, there is even less need to think about graph cloning/rewriting.

## Inference#

### MCMC#

The ability for model instance to generate conditional logp and dlogp function enable one of the unique feature of PyMC - CompoundStep method. On a conceptual level it is a Metropolis-within-Gibbs sampler. Users can specify different sampler for different RVs. Alternatively, it is implemented as yet another interceptor: The pm.sample(...) call will try to assign the best step methods to different free_RVs (e.g., NUTS if all free_RVs are continous). Then, (conditional) logp function(s) are compiled, and the sampler called each sampler within the list of CompoundStep in a for-loop for one sample circle.

For each sampler, it implements a step.step method to perform MH updates. Each time a dictionary (point in PyMC land, same structure as model.initial_point) is passed as input and output a new dictionary with the free_RVs being sampled now has a new value (if accepted, see here and here). There are some example in the CompoundStep doc:

#### Transition kernel#

The base class for most MCMC sampler (except SMC) is in ArrayStep. You can see that the step.step() is mapping the point into an array, and call self.astep(), which is an array in, array out function. A PyMC model compiles a conditional logp/dlogp function that replace the input RVs with a shared 1D tensor (flatten and stack view of the original RVs). And the transition kernel (i.e., .astep()) takes an array as input and output an array. See for example in the MH sampler.

This is of course very different compare to the transition kernel in e.g. TFP, which is a tenor in tensor out function. Moreover, transition kernels in TFP do not flatten the tensors, see eg docstring of tensorflow_probability/python/mcmc/random_walk_metropolis.py:

        new_state_fn: Python callable which takes a list of state parts and a
seed; returns a same-type list of Tensors, each being a perturbation
of the input state parts. The perturbation distribution is assumed to be
a symmetric distribution centered at the input state part.
Default value: None which is mapped to
tfp.mcmc.random_walk_normal_fn().


#### Dynamic HMC#

We love NUTS, or to be more precise Dynamic HMC with complex stopping rules. This part is actually all done outside of Aesara, for NUTS, it includes: The leapfrog, dual averaging, tunning of mass matrix and step size, the tree building, sampler related statistics like divergence and energy checking. We actually have an Aesara version of HMC, but it has never been used, and has been removed from the main repository. It can still be found in the git history, though.

#### Variational Inference (VI)#

The design of the VI module takes a different approach than MCMC - it has a functional design, and everything is done within Aesara (i.e., Optimization and building the variational objective). The base class of variational inference is pymc.variational.Inference, where it builds the objective function by calling:

...
self.objective = op(approx, **kwargs)(tf)
...


Where:

op     : Operator class
approx : Approximation class or instance
tf     : TestFunction instance
kwargs { kwargs passed to :class}Operator


The design is inspired by the great work Operator Variational Inference. Inference object is a very high level of VI implementation. It uses primitives: Operator, Approximation, and Test functions to combine them into single objective function. Currently we do not care too much about the test function, it is usually not required (and not implemented). The other primitives are defined as base classes in this file. We use inheritance to easily implement a broad class of VI methods leaving a lot of flexibility for further extensions.

For example, consider ADVI. We know that in the high-level, we are approximating the posterior in the latent space with a diagonal Multivariate Gaussian. In another word, we are approximating each elements in model.free_RVs with a Gaussian. Below is what happen in the set up:

def __init__(self, *args, **kwargs):
# ==> In the super class KLqp
super(KLqp, self).__init__(KL, MeanField(*args, **kwargs), None, beta=beta)
# ==> In the super class Inference
...
self.objective = KL(MeanField(*args, **kwargs))(None)
...


where KL is Operator based on Kullback Leibler Divergence (it does not need any test function).

...
def apply(self, f):
return -self.datalogp_norm + self.beta * (self.logq_norm - self.varlogp_norm)


Since the logp and logq are from the approximation, let’s dive in further on it (there is another abstraction here - Group - that allows you to combine approximation into new approximation, but we will skip this for now and only consider SingleGroupApproximation like MeanField): The definition of datalogp_norm, logq_norm, varlogp_norm are in variational/opvi, strip away the normalizing term, datalogp and varlogp are expectation of the variational free_RVs and data logp - we clone the datalogp and varlogp from the model, replace its input with Aesara tensor that samples from the variational posterior. For ADVI, these samples are from a Gaussian. Note that the samples from the posterior approximations are usually 1 dimension more, so that we can compute the expectation and get the gradient of the expectation (by computing the expectation of the gradient!). As for the logq since it is a Gaussian it is pretty straightforward to evaluate.

##### Some challenges and insights from implementing VI.#
• Graph based approach was helpful, but Aesara had no direct access to previously created nodes in the computational graph. You can find a lot of @node_property usages in implementation. This is done to cache nodes. TensorFlow has graph utils for that that could potentially help in doing this. On the other hand graph management in Tensorflow seemed to more tricky than expected. The high level reason is that graph is an add only container.

• There were few fixed bugs not obvoius in the first place. Aesara has a tool to manipulate the graph (aesara.clone_replace) and this tool requires extremely careful treatment when doing a lot of graph replacements at different level.

• We coined a term aesara.clone_replace curse. We got extremely dependent on this feature. Internal usages are uncountable:

• We use this to vectorize the model for both MCMC and VI to speed up computations

• We use this to create sampling graph for VI. This is the case you want posterior predictive as a part of computational graph.

As this is the core of the VI process, we were trying to replicate this pattern in TF. However, when aesara.clone_replace is called, Aesara creates a new part of the graph that can be collected by garbage collector, but TF’s graph is add only. So we should solve the problem of replacing input in a different way.

### Forward sampling#

As explained above, in distribution we have method to walk the model dependence graph and generate forward random sample in scipy/numpy. This allows us to do prior predictive samples using pymc.sampling.sample_prior_predictive see code. It is a fairly fast batch operation, but we have quite a lot of bugs and edge case especially in high dimensions. The biggest pain point is the automatic broadcasting. As in the batch random generation, we want to generate (n_sample, ) + RV.shape random samples. In some cases, where we broadcast RV1 and RV2 to create a RV3 that has one more batch shape, we get error (even worse, wrong answer with silent error).

The good news is, we are fixing these errors with the amazing works from lucianopaz and others. The challenge and some summary of the solution could be found in Luciano’s blog post

with pm.Model() as m:
mu = pm.Normal('mu', 0., 1., shape=(5, 1))
sigma = pm.HalfNormal('sigma', 5., shape=(1, 10))
pm.Normal('x', mu=mu, sigma=sigma, observed=np.random.randn(2, 5, 10))
trace = pm.sample_prior_predictive(100)

trace['x'].shape # ==> should be (100, 2, 5, 10)

pm.Normal.dist(mu=np.zeros(2), sigma=1).random(size=(10, 4))


There are also other error related random sample generation (e.g., Mixture is currently broken).

### Extending PyMC#

• Custom Inference method

• Connecting it to other library within a model

• Using “black box” likelihood function by creating a custom Aesara Op.

• Using emcee

• Using other library for inference

• Connecting to Julia for solving ODE (with gradient for solution that can be used in NUTS)

### What we got wrong#

#### Shape#

One of the pain point we often face is the issue of shape. The approach in TFP and pyro is currently much more rigorous. Adrian’s PR (https://github.com/pymc-devs/pymc/pull/2833) might fix this problem, but likely it is a huge effort of refactoring. I implemented quite a lot of patches for mixture distribution, but still they are not done very naturally.

#### Random methods in numpy#

There is a lot of complex logic for sampling from random variables, and because it is all in Python, we can’t transform a sampling graph further. Unfortunately, Aesara does not have code to sample from various distributions and we didn’t want to write that our own.

#### Samplers are in Python#

While having the samplers be written in Python allows for a lot of flexibility and intuitive for experiment (writing e.g. NUTS in Aesara is also very difficult), it comes at a performance penalty and makes sampling on the GPU very inefficient because memory needs to be copied for every logp evaluation.