class

ExponentialFamily

extendsDistribution

ExponentialFamily(batch_shape: tuple[int, ...] = (), event_shape: tuple[int, ...] = (), validate_args: bool | None = None)

source

Mix-in for exponential-family distributions.

An exponential-family distribution has a density of the form

p(x \mid \eta) = h(x)\, \exp\!\bigl(\eta^\top T(x) - A(\eta)\bigr)

where:

$\eta$ are the natural parameters (also called canonical parameters),
$T(x)$ is the vector of sufficient statistics,
$A(\eta) = \log Z(\eta)$ is the log-partition function (log-normaliser), and
$h(x)$ is the base measure.

This family encompasses most common distributions: Normal, Bernoulli, Gamma, Beta, Poisson, Dirichlet, and many more.

Subclasses provide _natural_params and _log_normalizer so that entropy can be derived from the standard exponential-family identity

H[X] = A(\eta) - \eta^\top \nabla_\eta A(\eta)

Per-distribution overrides of Distribution.entropy remain available for closed-form efficiency.

Notes

The exponential family structure implies:

The log-normalizer $A(\eta)$ is convex in $\eta$ .
$\nabla_\eta A(\eta) = \mathbb{E}[T(X)]$ (the mean parameter).
The Hessian $\nabla^2_\eta A(\eta)$ equals the covariance matrix of $T(X)$ , hence is always positive semi-definite.

These properties underpin many elegant results in information geometry, variational inference, and natural gradient methods.

Members of this family in lucid.distributions include lucid.distributions.Normal, lucid.distributions.Bernoulli, lucid.distributions.Categorical, lucid.distributions.Gamma, lucid.distributions.Beta, lucid.distributions.Dirichlet, lucid.distributions.Exponential, and lucid.distributions.Poisson. Heavy-tailed families such as lucid.distributions.Cauchy and Student's-t are not members.

Examples

>>> import lucid
>>> from lucid.distributions import Bernoulli
>>> d = Bernoulli(probs=0.7)  # ExponentialFamily subclass
>>> isinstance(d, lucid.distributions.distribution.ExponentialFamily)
True
>>> d.entropy()
Tensor(...)