Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 43 / 4

A credibility estimate combines a risk's observed experience with collective or prior information, typically as $Z\overline{X}+(1-Z)m_0$. The credibility factor $Z\in [0,1]$ is the weight assigned to the individual experience: more reliable or more extensive data generally increases it. Bayesian credibility treats the risk parameter as random with a prior distribution and predicts future claims by averaging the conditional claim mean under its posterior distribution. Under squared-error loss this posterior predictive mean is optimal. It need not be an affine combination in every Bayesian model; the Poisson-gamma conjugacy calculation below yields that credibility form exactly, rather than as a best-linear approximation.

Let $s={\sum }_{j=1}^n{x}_j$. The conditional joint likelihood function of the observed counts isThe prior is a gamma distribution with shape two and rate $\lambda$. Multiplication by its density gives the unnormalized posterior density ${\theta }^{s+1}{e}^{-(n+\lambda )\theta }$. Normalizing with the gamma function yieldsThus Poisson-gamma conjugacy gives $\Theta \mid \mathbf{x}\sim \mathrm{Gamma}(s+2,n+\lambda )$. Conditional on $\Theta$ and the past data, the next count has mean $\Theta$, so the law of total expectation givesWith $\overline{x}=s/n$ and prior mean $m_0=2/\lambda$, this is exactlyHere $n\ge 1$, $\lambda >0$, and the credibility factor increases to one as the number of observed years grows. The denominator uses the prior rate $\lambda$ and total exposure $n$; it does not add two to the exposure when adding the prior shape.

Given $\Theta =\theta$, the sum of the independent yearly counts is $S_n\sim \mathrm{Poisson}(n\theta )$: their probability generating functions multiply to $\exp (n\theta (z-1))$. Averaging this conditional law over the prior gives the mixed Poisson distributionThe likelihood for the full vector depends on $\theta$ only through $s$, so $S_n$ is a sufficient statistic; conditioning on the total or on the full observed vector gives the same posterior for $\Theta$.

For an event of positive probability $p_n(s)>0$, the posterior density is proportional to $e^{-n\theta }{\theta }^s\pi (\theta )$. Put $I_s={\int }_0^{\infty }{e}^{-n\theta }{\theta }^s\pi (\theta )d\theta$. Its mean is $I_{s+1}/I_s$. The future count has conditional mean $\Theta$, soThis is the posterior mean from adjacent mixed Poisson probabilities. Both probabilities refer to the same $n$-year exposure; the numerator is not a probability for $S_{n+1}$. For a proper prior and $n>0$, the integrals defining this posterior mean are finite because ${\theta }^{s+1}{e}^{-n\theta }$ is bounded on $[0,\infty )$.

Insert the shape-two gamma prior into the integral from part (ii). The gamma function integral givesTherefore $S_n$ has a negative binomial distribution with shape two and success probability $\lambda /(n+\lambda )$, using the failures-before-two-successes convention. This is the Poisson-gamma mixture. The identity ${\sum }_{s\ge 0}(s+1)z^s=(1-z{)}^{-2}$ verifies that the probabilities sum to one.

Its adjacent probability ratio isSubstituting in the posterior formula givesexactly the estimate from part (i). The equality is required by sufficiency, and the direct computation confirms that the two negative-binomial and gamma parameter conventions have been used consistently.