class

Poisson

extendsExponentialFamily

Poisson(rate: Tensor | float, validate_args: bool | None = None)

source

Poisson distribution over the non-negative integers.

Models the number of events occurring in a fixed interval when events happen independently at a constant mean rate $\lambda$ . The Poisson distribution is the limiting case of a Binomial as $n \to \infty$ and $p \to 0$ with $np = \lambda$ fixed.

Parameters

rateTensor | float

Mean rate

\lambda > 0

. Determines both the mean and the variance of the distribution.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Attributes

rateTensor

The mean rate parameter

\lambda

Notes

PMF (probability mass function):

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k \in \{0, 1, 2, \ldots\}

Moments:

Mean: $E[X] = \lambda$
Variance: $\operatorname{Var}[X] = \lambda$
Mode: $\lfloor \lambda \rfloor$

The log-probability is computed as $k \log \lambda - \lambda - \log\Gamma(k+1)$ , which is numerically stable via the lgamma function.

The Poisson has no closed-form entropy (it is an infinite sum over all non-negative integers), so entropy raises NotImplementedError.

Examples

>>> import lucid
>>> from lucid.distributions import Poisson
>>> dist = Poisson(rate=3.5)
>>> samples = dist.sample((100,))
>>> samples.shape
(100,)
>>> # log-probability at k=4
>>> dist.log_prob(lucid.tensor(4.0))

Methods (7)

dunder

init

→None

__init__(rate: Tensor | float, validate_args: bool | None = None)

source

Initialise a Poisson distribution.

Parameters

rateTensor | float

Mean rate parameter

\lambda > 0

. Determines both the mean and the variance.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

prop

mean

→Tensor

mean: Tensor

source

Mean of the Poisson distribution: $E[X] = \lambda$ .

Returns

Tensor

Mean values equal to rate, shape batch_shape.

prop

mode

→Tensor

mode: Tensor

source

Mode of the Poisson distribution: $\lfloor \lambda \rfloor$ .

When $\lambda$ is an integer, both $\lambda$ and $\lambda - 1$ are modes; this property returns $\lfloor \lambda \rfloor$ .

Returns

Tensor

Mode values of shape batch_shape.

prop

variance

→Tensor

variance: Tensor

source

Variance of the Poisson distribution: $\operatorname{Var}[X] = \lambda$ .

Returns

Tensor

Variance values equal to rate, shape batch_shape.

sample

→Tensor

sample(sample_shape: tuple[int, ...] = ())

source

Draw samples from the Poisson distribution.

Delegates to lucid.poisson, which uses the Knuth exact algorithm for small $\lambda$ and a Normal approximation with rounding for large $\lambda$ .

Parameters

sample_shapetuple[int, ...]= ()

Leading shape of the output sample batch.

Returns

Tensor

Non-negative integer samples of shape (*sample_shape, *batch_shape).

log_prob

→Tensor

log_prob(value: Tensor)

source

Log-probability of the given counts under the Poisson distribution.

\log P(X = k) = k \log \lambda - \lambda - \log\Gamma(k + 1)

Parameters

valueTensor

Non-negative integer counts

k \geq 0

Returns

Tensor

Log-probability values of the same shape as value.

entropy

→Tensor

entropy()

source

Entropy of the Poisson distribution (not available in closed form).

The Poisson entropy is an infinite sum over all non-negative integers and has no simple closed form.

Raises

NotImplementedError

Always — use numerical approximation if needed.