Exponential families for JAX
Project description
This library provides a set of tools for working with exponential family distributions in the differential programming library JAX.
The exponential families are an important class of probability distributions that include the normal, gamma, beta, exponential, Poisson, binomial, and Bernoulli distributions. For an explanation of the fundamental ideas behind this library, see our overview on exponential families.
The main motivation for using EFAX over a library like tensorflow-probability or the basic functions in JAX is that EFAX provides the two most important parametrizations for each exponential family—the natural and expectation parametrizations—and a uniform interface to efficient implementations of the main functions used in machine learning. An example of why this matters is that the most efficient way to implement cross entropy between X and Y relies on X being in the expectation parametrization and Y in the natural parametrization.
Framework
Representation
EFAX has a single base class for its objects: Parametrization whose type encodes the
distribution family.
Each parametrization object has a shape, and so it can store any number of distributions. Operations on these objects are vectorized. This is unlike SciPy where each distribution is represented by a single object, and so a thousand distributions need a thousand objects, and corresponding calls to functions that operate on them.
All parametrization objects are dataclasses using tjax.dataclass. These dataclasses are
a modification of Python’s dataclasses to support JAX’s “PyTree” type registration.
Each of the fields of a parametrization object stores a parameter over a specified support. Some parameters are marked as “fixed”, which means that they are fixed with respect to the exponential family. An example of a fixed parameter is the failure number of the negative binomial distribution.
For example:
@dataclass
class MultivariateNormalNP(NaturalParametrization['MultivariateNormalEP']):
mean_times_precision: RealArray = distribution_parameter(VectorSupport())
negative_half_precision: RealArray = distribution_parameter(SymmetricMatrixSupport())In this case, we see that there are two natural parameters for the multivariate normal distribution.
Objects of this type can hold any number of distributions: if such an object x has shape
s, then the shape of
x.mean_times_precision is (*s, n) and the shape of
x.negative_half_precision is (*s, n, n).
Parametrizations
Each exponential family distribution has two special parametrizations: the natural and the
expectation parametrization. (These are described in the overview pdf.)
Consequently, every distribution has at least two base classes, one inheriting from
NaturalParametrization and one from ExpectationParametrization.
The motivation for the natural parametrization is combining and scaling independent predictive evidence. In the natural parametrization, these operations correspond to scaling and addition.
The motivation for the expectation parametrization is combining independent observations into the maximum likelihood distribution that could have produced them. In the expectation parametrization, this is an expected value.
EFAX provides conversions between the two parametrizations through the
NaturalParametrization.to_exp and ExpectationParametrization.to_nat methods.
Important methods
EFAX aims to provide the main methods used in machine learning.
Every Parametrization has methods:
flattenedandunflattenedto flatten and unflatten the parameters into a single array. Typically, array-valued signals in a machine learning model would be unflattened into a distribution object, operated on, and then flattened before being sent back to the model. Flattening is careful with distributions with symmetric (or Hermitian) matrix-valued parameters. It only stores the upper triangular elements. And,shape, which supports broadcasting.
Every NaturalParametrization has methods:
to_expto convert itself to expectation parameters.sufficient_statisticsto produce the sufficient statistics given an observation (used in maximum likelihood estimation),pdfandlog_pdf, which is the density or mass function and its logarithm,fisher_information, which is the Fisher information matrix, andkl_divergence, which is the KL divergence.
Every ExpectationParametrization has methods:
to_natto convert itself to natural parameters, andkl_divergence, which is the KL divergence.
Some parametrizations inherit from these interfaces:
HasConjugatePriorcan produce the conjugate prior,HasGeneralizedConjugatePriorcan produce a generalization of the conjugate prior,Multidimensionaldistributions have a integer number of dimensions, andSamplabledistributions support sampling.
Some parametrizations inherit from these public mixins:
HasEntropyEPis an expectation parametrization with an entropy and cross entropy, andHasEntropyNPis a natural parametrization with an entropy, (The cross entropy is not efficient.)
Some parametrizations inherit from these private mixins:
ExpToNatimplements the conversion from expectation to natural parameters when no analytical solution is possible. It uses Newton’s method with a Jacobian to invert the gradient log-normalizer.TransformedNaturalParametrizationproduces a natural parametrization by relating it to an existing natural parametrization. And similarly forTransformedExpectationParametrization.
Distributions
EFAX supports the following distributions:
normal:
univariate real:
with fixed unit variance
with arbitrary variance
univariate complex
with fixed unit variance and zero pseudo-variance
with arbitrary variance
multivariate real:
with fixed unit variance
with fixed variance
with isotropic variance
with diagonal variance
with arbitrary variance
multivariate complex:
with fixed unit variance and zero pseudo-variance
circularly symmetric
on a finite set:
Bernoulli
multinomial
on the nonnegative integers:
geometric
logarithmic
negative binomial
Poisson
on the positive reals:
chi
chi-square
exponential
gamma
Rayleigh
Weibull
on the simplex:
beta
Dirichlet
generalized Dirichlet
on the n-sphere:
von Mises-Fisher
Usage
Basic usage
A basic use of the two parametrizations:
from jax import numpy as jnp
from efax import BernoulliEP, BernoulliNP
# p is the expectation parameters of three Bernoulli distributions having probabilities 0.4, 0.5,
# and 0.6.
p = BernoulliEP(jnp.array([0.4, 0.5, 0.6]))
# q is the natural parameters of three Bernoulli distributions having log-odds 0, which is
# probability 0.5.
q = BernoulliNP(jnp.zeros(3))
print(p.cross_entropy(q))
# [0.6931472 0.6931472 0.6931472]
# q2 is natural parameters of Bernoulli distributions having a probability of 0.3.
p2 = BernoulliEP(0.3 * jnp.ones(3))
q2 = p2.to_nat()
print(p.cross_entropy(q2))
# [0.6955941 0.78032386 0.86505365]
# A Bernoulli distribution with probability 0.3 predicts a Bernoulli observation with probability
# 0.4 better than the other observations.Optimization
Using the cross entropy to iteratively optimize a prediction is simple:
from __future__ import annotations
import jax.numpy as jnp
from jax import grad, jit, lax
from jax.tree_util import tree_map, tree_reduce
from tjax import BooleanNumeric, RealArray, RealNumeric, print_generic
from efax import BernoulliEP, BernoulliNP
def cross_entropy_loss(p: BernoulliEP, q: BernoulliNP) -> RealNumeric:
return jnp.sum(p.cross_entropy(q))
gce = jit(grad(cross_entropy_loss, 1))
def apply(x: RealArray, x_bar: RealArray) -> RealArray:
return x - 1e-4 * x_bar
def body_fun(q: BernoulliNP) -> BernoulliNP:
q_bar = gce(some_p, q)
return tree_map(apply, q, q_bar)
def cond_fun(q: BernoulliNP) -> BooleanNumeric:
q_bar = gce(some_p, q)
total = tree_reduce(jnp.sum,
tree_map(lambda x: jnp.sum(jnp.square(x)), q_bar))
return total > 1e-6
# some_p are expectation parameters of a Bernoulli distribution corresponding
# to probabilities 0.3, 0.4, and 0.7.
some_p = BernoulliEP(jnp.array([0.3, 0.4, 0.7]))
# some_q are natural parameters of a Bernoulli distribution corresponding to
# log-odds 0, which is probability 0.5.
some_q = BernoulliNP(jnp.zeros(3))
# Optimize the predictive distribution iteratively, and output the natural parameters of the
# prediction.
optimized_q = lax.while_loop(cond_fun, body_fun, some_q)
print_generic(optimized_q)
# BernoulliNP
# └── log_odds=Jax Array (3,) float32
# └── -0.8440 │ -0.4047 │ 0.8440
# Compare with the true value.
print_generic(some_p.to_nat())
# BernoulliNP
# └── log_odds=Jax Array (3,) float32
# └── -0.8473 │ -0.4055 │ 0.8473
# Print optimized natural parameters as expectation parameters.
print_generic(optimized_q.to_exp())
# BernoulliEP
# └── probability=Jax Array (3,) float32
# └── 0.3007 │ 0.4002 │ 0.6993Maximum likelihood estimation
Maximum likelihood estimation is often using the conjugate prior, but this can be done using only the expectation parametrization (which is equivalent less one parameter that represents the number of samples).
from functools import partial
import jax.numpy as jnp
from jax.random import key
from jax.tree_util import tree_map
from efax import DirichletNP
# Consider a Dirichlet distribution with a given alpha.
alpha = jnp.asarray([2.0, 3.0, 4.0])
source_distribution = DirichletNP(alpha_minus_one=alpha - 1.0)
# Let's sample from it.
n_samples = 10000
key_a = key(123)
samples = source_distribution.sample(key_a, (n_samples,))
# Now, let's find the maximum likelihood Dirichlet distribution that fits it.
# First, convert the samples to their sufficient statistics.
ss = DirichletNP.sufficient_statistics(samples)
# ss has type DirichletEP. This is similar to the conjguate prior of the Dirichlet distribution.
# Take the mean over the first axis.
ss_mean = tree_map(partial(jnp.mean, axis=0), ss) # ss_mean also has type DirichletEP.
# Convert this back to the natural parametrization.
estimated_distribution = ss_mean.to_nat()
print(estimated_distribution.alpha_minus_one + 1.0) # [1.9849904 3.0065458 3.963935 ]Contribution guidelines
Contributions are welcome!
It’s not hard to add a new distribution. The steps are:
Create an issue for the new distribution.
Solve for or research the equations needed to fill the blanks in the overview pdf, and put them in the issue. I’ll add them to the pdf for you.
Implement the natural and expectation parametrizations, either:
directly like in the Bernoulli distribution, or
as a transformation of an existing exponential family like the Rayleigh distribution.
Implement the conversion from the expectation to the natural parametrization. If this has no analytical solution, then there’s a mixin that implements a numerical solution. This can be seen in the Dirichlet distribution.
Add the new distribution to the tests by adding it to create_info.
Implementation should respect PEP8.
The tests can be run using pytest . -n auto. Specific distributions can be run with
pytest . -n auto --distribution=Gamma where the names match the class names in
create_info.
There are a few tools to clean and check the source:
ruff .pyrightmypyisort .pylint efax tests
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