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 explaination of the fundamental ideas behind this library, see our overview on exponential families.

## Installation

efax depends on https://github.com/google/jax/pull/2487, so the easiest way to get it working is to install jax from https://github.com/NeilGirdhar/jax:

```pip install --upgrade git+https://github.com/NeilGirdhar/jax.git
```

## Usage

In SciPy, a distribution is represented by a single object, so a thousand distributions need a thousand objects. Each object encodes the distribution family, and the parameters of the distribution. EFAX has a different representation. Each `ExponentialFamily` object encodes only the distribution family for many (say, one thousand) distributions. The parameters of the distributions are passed in to various methods on the object to evaluate various things. For example,

```from jax import numpy as jnp

from efax import Bernoulli

b = Bernoulli(shape=(3,))

# p are expectation parameters of Bernoulli distributions having probabilities
# 0.4, 0.5, and 0.6.
p = jnp.array([[0.4], [0.5], [0.6]])

# q are natural parameters of Bernoulli distributions having log-odds 0, which
# is probability 0.5.
q = jnp.zeros((3, 1))

print(b.cross_entropy(p, q))
# [0.6931472 0.6931472 0.6931472]

# q are natural parameters of Bernoulli distributions having a probability
# of 0.3.
q = b.exp_to_nat(0.3 * jnp.ones((3, 1)))

print(b.cross_entropy(p, q))
# [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.
```

With exponential families, maximum likelihood estimation is just expectation over expectation parameters. Models that combine independent predictors just sum natural parameters. When we want to optimize such models, we just want to take the gradient of cross entropy with respect to predictions.

Thanks to JAX, any gradient of the cross entropy will automatically be as accurate and numerically stable as possible. This is because the gradient of the cross entropy involves the gradient of the log-normalizer, which typically has a very nice form. For example,

```from jax import grad, jit, lax
from jax import numpy as jnp

from efax import Bernoulli

b = Bernoulli(shape=())

def cross_entropy_loss(p, q):
return b.cross_entropy(p, q)

def body_fun(q):
return q - gce(some_p, q) * 1e-4

def cond_fun(q):
return jnp.sum(gce(some_p, q) ** 2) > 1e-7

# some_p are expectation parameters of a Bernoulli distribution corresponding
# to probability 0.4.
some_p = jnp.array([0.4])

# some_q are natural parameters of a Bernoulli distribution corresponding to
# log-odds 0, which is probability 0.5.
some_q = jnp.array([0.0])

# Optimize the predictive distribution iteratively.
print(lax.while_loop(cond_fun, body_fun, some_q))
# Outputs the natural parameters that correspond to 0.4.

# Compare with the true value.
print(b.exp_to_nat(some_p))
```

## Contribution guidelines

• Conventions: PEP8.
• How to run tests: `pytest .`
• How to clean the source:
• `isort -rc -y`
• `pylint efax`
• `flake8 efax`
• `mypy efax`

## Project details

This version 0.2.2 0.2.1 0.2 0.1