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.

If $N\mid L$ has Poisson mean $L$, conditional expectation and the law of total variance give the display. A positive variance of the random mean prevents the unconditional count from being Poisson. For Poisson parents of density $\lambda$ and offspring mean $\mu$, the count in $A$ has excess variance $\lambda {\mu }^2\int \kappa (A-z{)}^{2}dz$, which is positive for a bounded positive-area set and a nontrivial model.

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.