Past exam of the statistics course of the University of Oxford / 2014 / 3 / b

Let $s={\sum }_i{x}_i$, $A=a+s$ and $B=b+n$. Conditional independence gives the Poisson distribution's likelihoodMultiply by the prior gamma distribution's density and discard factors independent of $\theta$:Normalizing with the gamma function identifiesHere $A=a+s$ is the shape and $B=b+n$ is the rate, not the scale. This is conjugate prior updating for a Poisson distribution.

The mean of a shape-rate gamma distribution is shape divided by rate. Hence the posterior mean isBoth weights are positive and sum to one. This is a convex combination of the prior mean and the sample mean, provingFor $n\ge 1$ it lies strictly between them if the two means differ, and equals both if they agree. The rate $b$ acts as the prior weight relative to the $n$ observed exposure units.

Let $F$ be the cumulative distribution function of a gamma distribution with shape $A=a+{\sum }_i{x}_i$ and rate $B=b+n$; explicitly,The posterior distribution is continuous and has positive density on $(0,\infty )$, so its quantiles give the requested equal-tailed credible interval:Equivalently, $2B\theta \mid x\sim {\chi }_{2A}^2$, so the endpoints can be written as the corresponding chi-squared distribution quantiles divided by $2B$. Either form excludes posterior probability $\alpha /2$ from each tail.

No. A permitted counterexample is $a=1$ with all observed counts zero. Then $A=1$ and the posterior distribution is an exponential distribution with rate $B=b+n$, whose density decreases strictly from the left boundary. Its equal-tailed credible interval isHere $L>0$. This excludes values just above zero whose posterior density is larger than that of retained values near $U$, violating the highest-density criterion. The actual highest posterior density interval starts at the support boundary and solves $1-e^{-Bh}=1-\alpha$:Writing zero as the endpoint causes no problem because a single point has zero posterior probability. Thus even within the gamma family the two intervals need not agree. More generally, for an interior unimodal gamma distribution, highest-density endpoints have equal densities, whereas equal-tailed endpoints need not.