Poissonian Likelihood#

This section shows how to evaluate a Poisson likelihood using derivkit.likelihood_kit.LikelihoodKit.

A Poisson likelihood describes the probability of observing discrete count data under a Poisson noise model with expected counts mu.

For observed integer counts data and a model prediction mu, the Poisson log-likelihood is

\[\ln p(\mathrm{data}\mid \mu) = \sum_i \left[ d_i \ln \mu_i - \mu_i - \ln(d_i!) \right].\]

Notation#

n denotes the number of count observations.
data contains n observed non-negative integer counts.
d_i denotes the observed count in the i-th observation.
mu is the expected count at each observation (model_parameters has shape (n,)).

The primary interface for evaluating the Poisson likelihood is derivkit.likelihood_kit.LikelihoodKit.poissonian(). For advanced usage, see derivkit.likelihoods.poisson.build_poisson_likelihood().

For a conceptual overview of likelihoods, see LikelihoodKit.

Poisson log-likelihood#

For inference, you should almost always work with the log-likelihood for numerical stability.

>>> import numpy as np
>>> from derivkit.likelihood_kit import LikelihoodKit
>>> # Observed counts (must be non-negative integers)
>>> counts = np.array([1, 2, 3, 4])
>>> # Expected counts (Poisson means)
>>> mu = np.array([0.8, 1.6, 2.4, 3.2])
>>> # Create LikelihoodKit instance
>>> lkit = LikelihoodKit(data=counts, model_parameters=mu)
>>> # Evaluate Poisson log-PMF
>>> counts_aligned, loglike = lkit.poissonian()
>>> print(np.size(loglike) == np.size(counts_aligned))
True
>>> print(np.all(np.isfinite(loglike)))
True
>>> print(np.array_equal(np.ravel(counts_aligned), counts))
True

Poisson PMF#

If you explicitly need probability mass values, set return_log=False.

>>> import numpy as np
>>> from derivkit.likelihood_kit import LikelihoodKit
>>> counts = np.array([1, 2, 3, 4])
>>> mu = np.array([0.8, 1.6, 2.4, 3.2])
>>> lkit = LikelihoodKit(data=counts, model_parameters=mu)
>>> counts_aligned, pmf = lkit.poissonian(return_log=False)
>>> print(np.array_equal(np.ravel(counts_aligned), counts))
True
>>> print(np.all((pmf >= 0.0) & np.isfinite(pmf)))
True
>>> print(np.size(pmf) == np.size(counts_aligned))
True

Log/linear consistency#

return_log=True and return_log=False are consistent up to exponentiation.

>>> import numpy as np
>>> from derivkit.likelihood_kit import LikelihoodKit
>>> counts = np.array([0, 1, 2, 3])
>>> mu = np.array([0.5, 0.8, 1.6, 2.4])
>>> lkit = LikelihoodKit(data=counts, model_parameters=mu)
>>> _, loglike = lkit.poissonian()
>>> _, pmf = lkit.poissonian(return_log=False)
>>> print(np.allclose(np.exp(loglike), pmf))
True

Notes#

model_parameters must provide one expected count per observation (mu has shape (n,)).
data must contain non-negative integers.
mu must be strictly positive to ensure a valid likelihood.
The Poisson likelihood assumes observations are conditionally independent.
By default, the Poisson likelihood returns the log-likelihood (return_log=True).
For large count values, working with the PMF directly can lead to numerical underflow; prefer log-likelihoods.
When combining multiple Poisson likelihood terms, sum log-likelihoods rather than multiplying PMFs.
For continuous data or approximately Gaussian noise, consider using a Gaussian likelihood instead.