= Solution
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.
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