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

Theorem: Let $y$ be a Poisson data set with exposure values. Suppose the event rate $\lambda$ has posterior distribution $\lambda \mid y \sim \mathrm{Gam}(a,b)$, i.e. a gamma distribution with density function

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

Let $t>0$ be a known future exposure and assume that a future observation satisfies $Y \mid \lambda,y \sim \mathrm{Poiss}(t\lambda)$. Then, for each integer $k \geq 0$, we have the probability mass function (PMF)

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

and the predictive moments of the future observation are

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

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

The distribution of $Y$ given $y$ is thus a negative binomial, with PMF on $k=0,1,2,\ldots$ in \eqref{eq:poiss-postpred-pmf} parametrized by shape $a$ and probability parameter $b/(b+t)$. The shape may be any positive real number.

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

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

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 $c=b+t$. Then, equation \eqref{eq:poiss-postpred-integral} becomes

\[\label{eq:poiss-postpred-integral-dev} \mathrm{Pr}(Y=k \mid y) = \frac{b^a t^k \Gamma(a+k)}{\Gamma(a) k! (b+t)^{a+k}}\]

which is equivalent to \eqref{eq:poiss-postpred-pmf}. It is a normalized PMF, because it mixes normalized Poisson mass functions with a normalized gamma density function. Exchanging the non-negative sum and integral preserves total probability one.

To obtain the posterior moments, multiply the gamma density \eqref{eq:poiss-postpred-gamma} 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 relation $\Gamma(z+1)=z\Gamma(z)$ 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 the laws of total expectation and total variance, now give:

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

This proves the PMF and both moments. The predictive count $Y$ has more variance than a Poisson count with the same mean. The term $at^2/b^2$ is the additional variance which comes from the posterior uncertainty about the event rate.

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

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