Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Count data ▷ Poisson distribution with exposure values ▷ Posterior predictive distribution

Theorem: Fix observed data $y$. Suppose the event rate has posterior $\lambda\mid y\sim\mathrm{Gam}(a,b)$ in shape-rate form, meaning

\[\label{eq:poiss-postpred-gamma} p(\lambda\mid y)=\frac{b^a}{\Gamma(a)}\lambda^{a-1}\exp(-b\lambda), \qquad \lambda>0,\quad a,b>0 \; .\]

Let $t>0$ be a known future exposure. Assume $Y_{\ast}\mid\lambda,y\sim\mathrm{Poiss}(t\lambda)$, with the future observation conditionally independent of the observed data given $\lambda$. For each integer $k\geq0$,

\[\label{eq:poiss-postpred-pmf} \mathrm{Pr}(Y_{\ast}=k\mid y) =\frac{\Gamma(a+k)}{\Gamma(a)k!} \left(\frac{b}{b+t}\right)^a \left(\frac{t}{b+t}\right)^k \; .\]

Its predictive moments are

\[\label{eq:poiss-postpred-moments} \mathrm{E}(Y_{\ast}\mid y)=\frac{at}{b},\qquad \mathrm{Var}(Y_{\ast}\mid y)=\frac{at}{b}+\frac{at^2}{b^2} \; .\]

The rate $\lambda$ is measured in events per exposure unit. The gamma rate parameter $b$ has the same exposure units as $t$, so $b+t$ and $t/b$ are well defined. The shape $a$ is dimensionless.

Here “negative binomial” means the PMF on $k=0,1,2,\ldots$ in \eqref{eq:poiss-postpred-pmf}, with shape $a$ and probability parameter $b/(b+t)$. The shape may be any positive real number. The PMF, rather than its name alone, fixes which convention is intended. The gamma scale used in the source is $1/b$ in this shape-rate convention.

Proof: Conditional on $\lambda$, the future PMF is $\exp(-t\lambda)(t\lambda)^k/k!$. By the definition of a posterior predictive distribution, average it over the gamma posterior:

\[\label{eq:poiss-postpred-integral} \begin{split} \mathrm{Pr}(Y_{\ast}=k\mid y) &=\int_0^\infty \frac{\exp(-t\lambda)(t\lambda)^k}{k!}\\ &\qquad{}\times\frac{b^a}{\Gamma(a)}\lambda^{a-1}\exp(-b\lambda) \,\mathrm{d}\lambda\\ &=\frac{b^at^k}{\Gamma(a)k!} \int_0^\infty \lambda^{a+k-1}\exp[-(b+t)\lambda] \,\mathrm{d}\lambda \; . \end{split}\]

For $s,c>0$, substituting $u=c\lambda$ in the defining gamma integral gives:

\[\label{eq:poiss-postpred-gamma-integral} \int_0^\infty\lambda^{s-1}\exp(-c\lambda)\,\mathrm{d}\lambda =\frac{\Gamma(s)}{c^s} \; .\]

Set $s=a+k$ and $c=b+t$. Equation \eqref{eq:poiss-postpred-integral} becomes $b^at^k\Gamma(a+k)/[\Gamma(a)k!(b+t)^{a+k}]$, which is \eqref{eq:poiss-postpred-pmf}. It is a normalized PMF because it mixes normalized Poisson PMFs against a normalized gamma density. Exchanging the nonnegative sum and integral preserves total probability one.

To obtain the posterior moments, multiply the gamma density by $\lambda^r$ and apply \eqref{eq:poiss-postpred-gamma-integral} with $s=a+r$ and $c=b$:

\[\label{eq:poiss-postpred-rate-moments} \begin{split} \mathrm{E}(\lambda^r\mid y) &=\frac{b^a}{\Gamma(a)} \int_0^\infty \lambda^{a+r-1}\exp(-b\lambda)\,\mathrm{d}\lambda\\ &=\frac{\Gamma(a+r)}{\Gamma(a)b^r},\qquad r=1,2 \; . \end{split}\]

Using the recurrence $\Gamma(z+1)=z\Gamma(z)$ gives $\mathrm{E}(\lambda\mid y)=a/b$ and $\mathrm{E}(\lambda^2\mid y)=a(a+1)/b^2$. The variance is the second moment minus the squared mean, so $\mathrm{Var}(\lambda\mid y)=a/b^2$. The Poisson mean and variance, together with total expectation and total variance, now give:

\[\label{eq:poiss-postpred-total-variance} \begin{split} \mathrm{E}(Y_{\ast}\mid y) &=\mathrm{E}[t\lambda\mid y]=\frac{at}{b},\\ \mathrm{Var}(Y_{\ast}\mid y) &=\mathrm{E}[\mathrm{Var}(Y_{\ast}\mid\lambda,y)\mid y] +\mathrm{Var}[\mathrm{E}(Y_{\ast}\mid\lambda,y)\mid y]\\ &=\mathrm{E}(t\lambda\mid y)+\mathrm{Var}(t\lambda\mid y) =\frac{at}{b}+\frac{at^2}{b^2} \; . \end{split}\]

This proves the PMF and both moments.

The predictive count has more variance than a Poisson count with the same mean. The term $at^2/b^2$ is the additional variance from posterior uncertainty about the event rate.

∎
Sources:
  • Gary Koop, Dale J. Poirier, Justin L. Tobias (2007): "Prediction"; in: Bayesian Econometric Methods, Exercise 7.2, pp. 73-74 (gamma shape-scale parameterization).

Metadata: ID: P558 | shortcut: poiss-postpred | author: kjpetrykowski | date: 2026-10-17, 21:51.