Apache 2.0

Distrax is a lightweight library of probability distributions and bijectors. It acts as a JAX-native reimplementation of a subset of TensorFlow Probability (TFP), with some new features and emphasis on extensibility.

Distrax can be installed with pip directly from GitHub:

`pip install git+git://github.com/deepmind/distrax.git`

.

or from PyPI:

`pip install distrax`

To run the tests or examples you will need to install additional requirements.

The general design principles for the DeepMind JAX Ecosystem are addressed in this blog. Additionally, Distrax places emphasis on the following:

**Readability.**Distrax implementations are intended to be self-contained and read as close to the underlying math as possible.**Extensibility.**We have made it as simple as possible for users to define their own distribution or bijector. This is useful for example in reinforcement learning, where users may wish to define custom behavior for probabilistic agent policies.**Compatibility.**Distrax is not intended as a replacement for TFP, and TFP contains many advanced features that we do not intend to replicate. To this end, we have made the APIs for distributions and bijectors as cross-compatible as possible, and provide utilities for transforming between equivalent Distrax and TFP classes.

Distributions in Distrax are simple to define and use, particularly if you're used to TFP. Let's compare the two side-by-side:

```
import distrax
import jax
import jax.numpy as jnp
from tensorflow_probability.substrates import jax as tfp
tfd = tfp.distributions
key = jax.random.PRNGKey(1234)
mu = jnp.array([-1., 0., 1.])
sigma = jnp.array([0.1, 0.2, 0.3])
dist_distrax = distrax.MultivariateNormalDiag(mu, sigma)
dist_tfp = tfd.MultivariateNormalDiag(mu, sigma)
samples = dist_distrax.sample(seed=key)
# Both print 1.775
print(dist_distrax.log_prob(samples))
print(dist_tfp.log_prob(samples))
```

In addition to behaving consistently, Distrax distributions and TFP distributions are cross-compatible. For example:

```
mu_0 = jnp.array([-1., 0., 1.])
sigma_0 = jnp.array([0.1, 0.2, 0.3])
dist_distrax = distrax.MultivariateNormalDiag(mu_0, sigma_0)
mu_1 = jnp.array([1., 2., 3.])
sigma_1 = jnp.array([0.2, 0.3, 0.4])
dist_tfp = tfd.MultivariateNormalDiag(mu_1, sigma_1)
# Both print 85.237
print(dist_distrax.kl_divergence(dist_tfp))
print(tfd.kl_divergence(dist_distrax, dist_tfp))
```

Distrax distributions implement the method `sample_and_log_prob`

, which provides
samples and their log-probability in one line. For some distributions, this is
more efficient than calling separately `sample`

and `log_prob`

:

```
mu = jnp.array([-1., 0., 1.])
sigma = jnp.array([0.1, 0.2, 0.3])
dist_distrax = distrax.MultivariateNormalDiag(mu, sigma)
samples = dist_distrax.sample(seed=key, sample_shape=())
log_prob = dist_distrax.log_prob(samples)
# A one-line equivalent of the above is:
samples, log_prob = dist_distrax.sample_and_log_prob(seed=key, sample_shape=())
```

TFP distributions can be passed to Distrax meta-distributions as inputs. For example:

```
key = jax.random.PRNGKey(1234)
mu = jnp.array([-1., 0., 1.])
sigma = jnp.array([0.2, 0.3, 0.4])
dist_tfp = tfd.Normal(mu, sigma)
metadist_distrax = distrax.Independent(dist_tfp, reinterpreted_batch_ndims=1)
samples = metadist_distrax.sample(seed=key)
print(metadist_distrax.log_prob(samples)) # Prints 0.38871175
```

To use Distrax distributions in TFP meta-distributions, Distrax provides the
wrapper `to_tfp`

. A wrapped Distrax distribution can be directly used in TFP:

```
key = jax.random.PRNGKey(1234)
distrax_dist = distrax.Normal(0., 1.)
wrapped_dist = distrax.to_tfp(distrax_dist)
metadist_tfp = tfd.Sample(wrapped_dist, sample_shape=[3])
samples = metadist_tfp.sample(seed=key)
print(metadist_tfp.log_prob(samples)) # Prints -3.3409896
```

A "bijector" in Distrax is an invertible function that knows how to compute its Jacobian determinant. Bijectors can be used to create complex distributions by transforming simpler ones. Distrax bijectors are functionally similar to TFP bijectors, with a few API differences. Here is an example comparing the two:

```
import distrax
import jax.numpy as jnp
from tensorflow_probability.substrates import jax as tfp
tfb = tfp.bijectors
tfd = tfp.distributions
# Same distribution.
distrax.Transformed(distrax.Normal(loc=0., scale=1.), distrax.Tanh())
tfd.TransformedDistribution(tfd.Normal(loc=0., scale=1.), tfb.Tanh())
```

Additionally, Distrax bijectors can be composed and inverted:

```
bij_distrax = distrax.Tanh()
bij_tfp = tfb.Tanh()
# Same bijector.
inv_bij_distrax = distrax.Inverse(bij_distrax)
inv_bij_tfp = tfb.Invert(bij_tfp)
# These are both the identity bijector.
distrax.Chain([bij_distrax, inv_bij_distrax])
tfb.Chain([bij_tfp, inv_bij_tfp])
```

All TFP bijectors can be passed to Distrax, and can be freely composed with Distrax bijectors. For example, all of the following will work:

```
distrax.Inverse(tfb.Tanh())
distrax.Chain([tfb.Tanh(), distrax.Tanh()])
distrax.Transformed(tfd.Normal(loc=0., scale=1.), tfb.Tanh())
```

Distrax bijectors can also be passed to TFP, but first they must be transformed
with `to_tfp`

:

```
bij_distrax = distrax.to_tfp(distrax.Tanh())
tfb.Invert(bij_distrax)
tfb.Chain([tfb.Tanh(), bij_distrax])
tfd.TransformedDistribution(tfd.Normal(loc=0., scale=1.), bij_distrax)
```

Distrax also comes with `Lambda`

, a convenient wrapper for turning simple JAX
functions into bijectors. Here are a few `Lambda`

examples with their TFP
equivalents:

```
distrax.Lambda(lambda x: x)
# tfb.Identity()
distrax.Lambda(lambda x: 2*x + 3)
# tfb.Chain([tfb.Shift(3), tfb.Scale(2)])
distrax.Lambda(jnp.sinh)
# tfb.Sinh()
distrax.Lambda(lambda x: jnp.sinh(2*x + 3))
# tfb.Chain([tfb.Sinh(), tfb.Shift(3), tfb.Scale(2)])
```

Unlike TFP, bijectors in Distrax do not take `event_ndims`

as an argument when
they compute the Jacobian determinant. Instead, Distrax assumes that the number
of event dimensions is statically known to every bijector, and uses
`Block`

to lift bijectors to a different number of dimensions. For example:

```
x = jnp.zeros([2, 3, 4])
# In TFP, `event_ndims` can be passed to the bijector.
bij_tfp = tfb.Tanh()
ld_1 = bij_tfp.forward_log_det_jacobian(x, event_ndims=0) # Shape = [2, 3, 4]
# Distrax assumes `Tanh` is a scalar bijector by default.
bij_distrax = distrax.Tanh()
ld_2 = bij_distrax.forward_log_det_jacobian(x) # ld_1 == ld_2
# With `event_ndims=2`, TFP sums the last 2 dimensions of the log det.
ld_3 = bij_tfp.forward_log_det_jacobian(x, event_ndims=2) # Shape = [2]
# Distrax treats the number of dimensions statically.
bij_distrax = distrax.Block(bij_distrax, ndims=2)
ld_4 = bij_distrax.forward_log_det_jacobian(x) # ld_3 == ld_4
```

Distrax bijectors implement the method `forward_and_log_det`

(some bijectors
additionally implement `inverse_and_log_det`

), which allows to obtain the
`forward`

mapping and its log Jacobian determinant in one line. For some
bijectors, this is more efficient than calling separately `forward`

and
`forward_log_det_jacobian`

. (Analogously, when available, `inverse_and_log_det`

can be more efficient than `inverse`

and `inverse_log_det_jacobian`

.)

```
x = jnp.zeros([2, 3, 4])
bij_distrax = distrax.Tanh()
y = bij_distrax.forward(x)
ld = bij_distrax.forward_log_det_jacobian(x)
# A one-line equivalent of the above is:
y, ld = bij_distrax.forward_and_log_det(x)
```

Distrax distributions and bijectors can be passed as arguments to jitted
functions. User-defined distributions and bijectors get this property for free
by subclassing `distrax.Distribution`

and `distrax.Bijector`

respectively. For
example:

```
mu_0 = jnp.array([-1., 0., 1.])
sigma_0 = jnp.array([0.1, 0.2, 0.3])
dist_0 = distrax.MultivariateNormalDiag(mu_0, sigma_0)
mu_1 = jnp.array([1., 2., 3.])
sigma_1 = jnp.array([0.2, 0.3, 0.4])
dist_1 = distrax.MultivariateNormalDiag(mu_1, sigma_1)
jitted_kl = jax.jit(lambda d_0, d_1: d_0.kl_divergence(d_1))
# Both print 85.237
print(jitted_kl(dist_0, dist_1))
print(dist_0.kl_divergence(dist_1))
```

User-defined distributions can be created by subclassing `distrax.Distribution`

.
This can be achieved by implementing only a few methods:

```
class MyDistribution(distrax.Distribution):
def __init__(self, ...):
...
def _sample_n(self, key, n):
samples = ...
return samples
def log_prob(self, value):
log_prob = ...
return log_prob
def event_shape(self):
event_shape = ...
return event_shape
def _sample_n_and_log_prob(self, key, n):
# Optional. Only when more efficient implementation is possible.
samples, log_prob = ...
return samples, log_prob
```

Similarly, more complicated bijectors can be created by subclassing
`distrax.Bijector`

. This can be achieved by implementing only one or two class
methods:

```
class MyBijector(distrax.Bijector):
def __init__(self, ...):
super().__init__(...)
def forward_and_log_det(self, x):
y = ...
logdet = ...
return y, logdet
def inverse_and_log_det(self, y):
# Optional. Can be omitted if inverse methods are not needed.
x = ...
logdet = ...
return x, logdet
```

The `examples`

directory contains some representative examples of full programs
that use Distrax.

`hmm.py`

demonstrates how to use `distrax.HMM`

to combine distributions that
model the initial states, transitions, and observation distributions of a
Hidden Markov Model, and infer the latent rates and state transitions in a
changing noisy signal.

`vae.py`

contains an example implementation of a variational auto-encoder that
is trained to model the binarized MNIST dataset as a joint `distrax.Bernoulli`

distribution over the pixels.

`flow.py`

illustrates a simple example of modelling MNIST data using
`distrax.MaskedCoupling`

layers to implement a normalizing flow, and training
the model with gradient descent.

We greatly appreciate the ongoing support of the TensorFlow Probability authors in assisting with the design and cross-compatibility of Distrax.

Special thanks to Aleyna Kara and Kevin Murphy for contributing the code upon which the Hidden Markov Model and associated example are based.

To cite this repository:

```
@software{distrax2021github,
author = {Jake Bruce and David Budden and Matteo Hessel and George Papamakarios and Francisco Ruiz},
title = {Distrax: Probability distributions in {JAX}},
url = {http://github.com/deepmind/distrax},
version = {0.0.1},
year = {2021},
}
```

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