Mixed Poisson distribution ($S\mid \Theta \sim \mathrm{Poisson}(n\Theta )$)

A mixture of Poisson distributions over a nonnegative random intensity. With exposure $n>0$ and prior density $\pi (\theta )$, its probability mass function is $p_n(s)={\int }_0^{\infty }{e}^{-n\theta }(n\theta {)}^s\pi (\theta )d\theta /s!$. A gamma mixing distribution gives a negative binomial distribution. Counts sharing the same latent intensity are conditionally independent, but generally dependent after mixing.

Let $L\ge 0$ be a finite random variable with finite second moment, and let $N\mid L$ have a Poisson distribution of mean $L$. Then conditional expectation and the law of total variance giveA nondegenerate mixing law has positive variance, so the count is overdispersed relative to a Poisson distribution with the same expected value. For example, equal mixing between means $1$ and $3$ gives count expected value $2$ and variance $3$. The finite-moment assumptions justify these ordinary variance identities; a proper mixing law can instead yield infinite count moments.

A spatial example uses Poisson point process parent locations of intensity $\lambda >0$, independent Poisson distribution offspring counts of expected value $\mu >0$, and an offspring displacement probability distribution $\kappa$. Given the parents, the count in a bounded measurable region $A$ is Poisson with random mean $L_A=\mu {\sum }_z\kappa (A-z)$. Its excess variance isThis follows from Campbell theorem. If $A$ has positive finite volume, the integral is finite and positive: $0\le \kappa (A-z)\le 1$ and its integral is the volume of $A$. Thus shared random parent locations produce extra count variation even though the offspring counts are conditionally Poisson.

For a mixed Poisson distribution with known exposure $n>0$, multiply the unnormalized posterior density ${\theta }^s{e}^{-n\theta }\pi (\theta )$ by $\theta$ and compare its integral with the next probability mass. The factors $n^s/s!$ give the displayed identity whenever $p_n(s)>0$. Both mass probabilities have the same exposure $n$. For a future conditionally independent unit-exposure Poisson distribution count, this is also its posterior predictive expected value. It holds for arbitrary proper mixing distributions, with integrals taken against the prior measure when there is no density.