# PyMC3 Developer Guide¶

PyMC3 is a Python package for Bayesian statistical modeling built on top of Theano. This document aims to explain the design and implementation of probabilistic programming in PyMC3, with comparisons to other PPL like TensorFlow Probability (TFP) and Pyro in mind. 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¶

A high-level introduction of Distribution in PyMC3 can be found in the documentation. The source code of the probability distributions is nested under pymc3/distributions, with the Distribution class defined in distribution.py. A few important points to highlight in the Distribution Class:

class Distribution:
"""Statistical distribution"""
def __new__(cls, name, *args, **kwargs):
...
try:
model = Model.get_context()
except TypeError:
raise TypeError(...

if isinstance(name, string_types):
...
dist = cls.dist(*args, **kwargs)
return model.Var(name, dist, ...)
...


In a way, the snippet above represents the unique features of pymc3’s Distribution class:

• Distribution objects are only usable inside of a Model context. If they are created outside of the model context manager, it raises an error.
• A Distribution requires at least a name argument, and other parameters that defines the Distribution.
• When a Distribution is initialized inside of a Model context, two things happen:
1. a stateless distribution is initialized dist = {DISTRIBUTION_cls}.dist(*args, **kwargs);
2. a random variable following the said distribution is added to the model model.Var(name, dist, ...)

Thus, users who are building models using with pm.Model() ... should be aware that they are never directly exposed to static and stateless distributions, but rather random variables that follow some density functions. Instead, to access a stateless distribution, you need to call pm.SomeDistribution.dist(...) or RV.dist after you initialized RV in a model context (see https://docs.pymc.io/prob_dists.html#using-pymc-distributions-without-a-model).

With this distinction in mind, we can take a closer look at the stateless distribution part of pymc3 (see distriution api in doc), which divided into:

• Continuous
• Discrete
• Multivariate
• Mixture
• Timeseries

Quote from the doc:

All distributions in pm.distributions will have two important methods: random() and logp() with the following signatures:
class SomeDistribution(Continuous):
def __init__(...):
...

def random(self, point=None, size=None):
...
return random_samples

def logp(self, value):
...


PyMC3 expects the logp() method to return a log-probability evaluated at the passed value argument. This method is used internally by all of the inference methods to calculate the model log-probability, which is then used for fitting models. The random() method is used to simulate values from the variable, and is used internally for posterior predictive checks.

In the PyMC3 Distribution class, the logp() method is the most elementary. As long as you have a well-behaved density function, we can use it in the model to build the model log-likelihood function. Random number generation is great to have, but sometimes there might not be efficient random number generator for some densities. Since a function is all you need, you can wrap almost any thenao function into a distribution using pm.DensityDist https://docs.pymc.io/prob_dists.html#custom-distributions

Thus, distributions that are defined in the distributions submodule (e.g. look at pm.Normal in pymc3.distributions.continuous), each describes a family of probabilistic distribution (no different from distribution in other PPL library). Once it is initialised within a model context, it contains properties that are related to the random variable (e.g. mean/expectation). Note that if the parameters are constants, these properties could be the same as the distribution properties.

### Reflection¶

How tensor/value semantics for probability distributions is enabled in pymc3:

In PyMC3, we treat x = Normal('x', 0, 1) as defining a random variable (intercepted and collected under a model context, more on that below), and x.dist() as the associated density/mass function (distribution in the mathematical sense). It is not perfect, and now after a few years learning Bayesian statistics I also realized these subtleties (i.e., the distinction between random variable and distribution). But when I was learning probabilistic modelling as a beginner, I did find this approach to be the easiest and most straightforward. In a perfect world, we should have $$x \sim \text{Normal}(0, 1)$$ which defines a random variable that follows a Gaussian distribution, and $$\chi = \text{Normal}(0, 1), x \sim \chi$$ which define a scalar density function that takes input $$x$$

(X:=f(x) = 1/sqrt(2*pi) * exp(-.5*x**2))

Within a model context, RVs are essentially Theano tensors (more on that below). This is different than TFP and pyro, where you need to be more explicit about the conversion. For example:

PyMC3

with pm.Model() as model:
z = pm.Normal('z', mu=0., sigma=5.)             # ==> pymc3.model.FreeRV, or theano.tensor with logp
x = pm.Normal('x', mu=z, sigma=1., observed=5.) # ==> pymc3.model.ObservedRV, also has logp properties
x.logp({'z': 2.5})                               # ==> -4.0439386
model.logp({'z': 2.5})                           # ==> -6.6973152


TFP

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
sess = tf.Session()
sess.run(x, feed_dict={z: 2.5})                  # ==> -4.0439386
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


### Random method and logp method, very different behind the curtain¶

In short, the random method is scipy/numpy-based, and the logp method is Theano-based. The logp method is straightforward - it is a Theano function within each distribution. It has the following signature:

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) theano tensor
total_log_prob = f(param1, param2, ..., value)


In the logp method, parameters and values are either Theano 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 totall logp) the dependence is taken care of by Theano 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 PyMC3 we use the second approach by using theano.clone() when it is needed.

As explained above, distribution in a pm.Model() context automatically turn into a tensor with distribution property (pymc3 random variable). To get the logp of a free_RV is just evaluating the logp() on itself:

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


Or for a ObservedRV. it evaluate the logp on the data:

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


However, for the random method things are a bit less graceful. As the random generator is limited in Theano, all random generation is done in scipy/numpy land. In the random method, we have:

def random(self, point=None, size=None):
# GET PARAMETERS
param1, param2, ... = draw_values([self.param1, self.param2, ...],
point=point,
size=size)
# GENERATE SAMPLE
samples = generate_samples(SCIPY_OR_NUMPY_RANDOM_FUNCTION,
param1, param2, ... # ==> parameters, type is numpy arrays
dist_shape=self.shape,
size=size)
return samples


Here, point is a dictionary that contains dependence of param1, param2, ..., and draw_values generates a (random) (size, ) + param.shape arrays conditioned on the information from point. This is the backbone for forwarding random simulation. The draw_values function is a recursive algorithm to try to resolve all the dependence outside of Theano, by walking the Theano computational graph, it is complicated and a constant pain point for bug fixing: https://github.com/pymc-devs/pymc3/blob/master/pymc3/distributions/distribution.py#L217-L529 (But also see a recent PR that use interception and context manager to resolve the dependence issue)

## Model context and Random Variable¶

I like to think that the with pm.Model() ... is a key syntax feature and the signature of PyMC3 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 eg 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.)


Which is the same as doing:

m = pm.Model()
x = m.Var('x', pm.Normal.dist(mu=0., sigma=1.))


Both with the same output:

print(type(x))                              # ==> <class 'pymc3.model.FreeRV'>
print(m.free_RVs)                           # ==> [x]
print(x.distribution.logp(5.))              # ==> Elemwise{switch,no_inplace}.0
print(x.distribution.logp(5.).eval({}))     # ==> -13.418938533204672
print(m.logp({'x': 5.}))                    # ==> -13.418938533204672


Looking closer to the classmethod model.Var, it is clear that what PyMC3 does is an interception of the Random Variable, depending on the *args: https://github.com/pymc-devs/pymc3/blob/6d07591962a6c135640a3c31903eba66b34e71d8/pymc3/model.py#L786-L847

def Var(self, name, dist, data=None, total_size=None):
"""
...
"""
...
if data is None:
if getattr(dist, "transform", None) is None:
with self:
var = FreeRV(...)             # ==> FreeRV
self.free_RVs.append(var)
else:
with self:
var = TransformedRV(...)      # ==> TransformedRV
...
self.deterministics.append(var)
return var
elif isinstance(data, dict):
with self:
var = MultiObservedRV(...)        # ==> MultiObservedRV
self.observed_RVs.append(var)
if var.missing_values:
...                               # ==> Additional FreeRV if there is missing values
else:
with self:
var = ObservedRV(...)             # ==> ObservedRV
self.observed_RVs.append(var)
if var.missing_values:
...                               # ==> Additional FreeRV if there is missing values

return var


In general, if there is observed, the RV is defined as a ObservedRV, otherwise if it has a transformed method, it is a TransformedRV, otherwise, it returns the most elementary form: a FreeRV.

Below, I will take a deeper look into TransformedRV, a normal user might not necessary come in contact with the concept, as TransformedRV and TransformedDistribution are intentionally not user facing.

Because in PyMC3 there is no bijector class like in TFP or pyro, we only have a partial implementation called Transform, which implements Jacobian correction for forward mapping only (there is no Jacobian correction for inverse mapping). The use case we considered are limited to the set of distributions that are bounded, and the transformation maps the bounded set to the real line - see doc. In general, PyMC3 does not provide explicit functionality to transform one distribution to another. Instead, a dedicated distribution is usually created in consideration of optimising performance. But getting a TransformedDistribution is also possible (see also in doc):

tr = pm.distributions.transforms
class Exp(tr.ElemwiseTransform):
name = "exp"
def backward(self, x):
return tt.log(x)
def forward(self, x):
return tt.exp(x)
def jacobian_det(self, x):
return -tt.log(x)

lognorm = Exp().apply(pm.Normal.dist(0., 1.))
lognorm

<pymc3.distributions.transforms.TransformedDistribution at 0x7f1536749b00>


Now, back to model.RV(...) - things return from model.RV(...) are Theano tensor variables, and it is clear from looking at TransformedRV:

class TransformedRV(TensorVariable):
...


as for FreeRV and ObservedRV, they are TensorVariable with Factor:

class FreeRV(Factor, TensorVariable):
...


and Factor basically enable and assign the logp (representated as a tensor also) property to a Theano tensor (thus making it a random variable). For a TransformedRV, it transform the distribution into a TransformedDistribution, and then model.Var is called again to added the RV associated with the TransformedDistribution as a FreeRV:

...
self.transformed = model.Var(
transformed_name, transform.apply(distribution), total_size=total_size)


note: after transform.apply(distribution) its .transform porperty is set to None, thus making sure that the above call will only add one FreeRV. In another word, you cannot do chain transformation by nested applying multiple transforms to a Distribution (however, you can use Chain transformation).

z = pm.Lognormal.dist(mu=0., sigma=1., transform=tr.Log)
z.transform           # ==> pymc3.distributions.transforms.Log

z2 = Exp().apply(z)
z2.transform is None  # ==> True


### 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/Theano ndarray, thus allowing logp/dlogp function to have two equivalent version: one take a dict as input and the other take a ndarray as input. More importantly, a pm.Model() contains methods to compile Theano function that takes Random Variables (that are also initialised within the same model) as input.

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

print(m.test_point)
print(m.dict_to_array(m.test_point))  # ==> m.bijection.map(m.test_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):
"""Theano scalar of log-probability of the model"""
with self:
factors = [var.logpt for var in self.basic_RVs] + self.potentials
logp = tt.sum([tt.sum(factor) for factor in factors])
...
return logp


which returns a Theano tensor that its value depends on the free parameters in the model (i.e., its parent nodes from the Theano 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 Theano graph, thus you cannot pass new tensor to generate a logp function. For similar reason, in PyMC3 we do graph copying a lot using theano.clone 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)         # ==> theano.tensor.var.TensorVariable

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


output:

array(-51.25369126)


PyMC3 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 Theano graph to compile additional Theano functions. PyMC3 relies on theano.clone to copy the model.logpt and replace its input. It does not edit or rewrite the graph directly.

class ValueGradFunction:
"""Create a theano function that computes a value and its gradient.
...
"""
def __init__(self, logpt, grad_vars, extra_vars=[], dtype=None,
casting='no', **kwargs):
...

self._extra_vars = extra_vars
self._extra_var_names = set(var.name for var in extra_vars)
self._logpt = logpt
self.size = self._ordering.size
self._extra_are_set = False

...

# Extra vars are a subset of free_RVs that are not input to the compiled function.
# But nonetheless logpt depends on these RVs.
# This is set up as a dict of theano.shared tensors, but givens (a list of
# tuple(free_RVs, theano.shared)) is the actual list that goes into the theano function
givens = []
self._extra_vars_shared = {}
for var in extra_vars:
shared = theano.shared(var.tag.test_value, var.name + '_shared__')
self._extra_vars_shared[var.name] = shared
givens.append((var, shared))

# See the implementation below. Basically, it clones the logpt and replaces its
# input with a *single* 1d theano tensor
self._vars_joined, self._logpt_joined = self._build_joined(

inputs = [self._vars_joined]

self._theano_function = theano.function(

def _build_joined(self, logpt, args, vmap):
args_joined = tt.vector('__args_joined')
args_joined.tag.test_value = np.zeros(self.size, dtype=self.dtype)

joined_slices = {}
for vmap in vmap:
sliced = args_joined[vmap.slc].reshape(vmap.shp)
sliced.name = vmap.var
joined_slices[vmap.var] = sliced

replace = {var: joined_slices[var.name] for var in args}
return args_joined, theano.clone(logpt, replace=replace)

...
logp, dlogp = self._theano_function(array)
return

def set_extra_values(self, extra_vars):
...

def get_extra_values(self):
...

@property
def profile(self):
...

def dict_to_array(self, point):
...

def array_to_dict(self, array):
...

def array_to_full_dict(self, array):
"""Convert an array to a dictionary with grad_vars and extra_vars."""
...

...


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 Theano directly:

import theano
func = theano.function(m.free_RVs, m.logpt)
func(*inputlist)

array(-51.0769075)

logpt_grad = theano.grad(m.logpt, m.free_RVs)
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 = theano.shared(inputlist[1])
func2 = theano.function([m.free_RVs[0]], m.logpt, givens=[(m.free_RVs[1], shared)])
print(func2(inputlist[0]))

func_d2 = theano.function([m.free_RVs[0]], logpt_grad2, givens=[(m.free_RVs[1], shared)])
print(func_d2(inputlist[0]))

-51.07690750130328
[ 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 = theano.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 theano.clone, we always have the input to the Theano 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.

### Reflection¶

The current setup is quite powerful, as the Theano 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 Theano 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, theano.clone 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 PyMC3 - CompoundStep method. It is conceptual level it is a Metropolis-within-Gibbs sampler. User can specify different sampler of 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 PyMC3 land, same structure as model.test_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 pymc3 model compile 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 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 eg 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 stoping rules. This part is actually all done outside of Theano, 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 a Theano version of HMC: https://github.com/pymc-devs/pymc3/blob/master/pymc3/step_methods/hmc/trajectory.py but it is never been used.

### 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 Theano (i.e., Optimization and building the variational objective). The base class of variational inference is pymc3.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 Inferece
...
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 Theano 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 Theano 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. Theano has a tool to manipulate the graph (theano.clone) and this tool requires extremely careful treatment when doing a lot of graph replacements at different level.
• We coined a term theano.clone 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 theano.clone is called, Theano 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 pymc3.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):

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

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

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


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

## Extending PyMC3¶

• Custom Inference method
• Connecting it to other library within a model
• 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/pymc3/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, Theano 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 Theano 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.