= Inflation-adjusted Poisson-gamma credibility
{title2=$\widehat Y_{n+1}=c_{n+1}\frac{\alpha+\sum_jw_jy_j/c_j}{\beta+\sum_jw_j}$}
For a common <gamma distribution> (shape $\alpha$, rate $\beta$) frequency <prior distribution> and conditionally <independent> <Poisson distribution> yearly counts of means $w_j\Theta$, observed average amounts $y_j$ determine counts $k_j=w_jy_j/c_j$. The <posterior distribution> shape is $\alpha+\sum_jk_j$ and rate is $\beta+\sum_jw_j$. The future per-policy <mean> is the display, equal to <credibility factor> $W/(W+\beta)$ times $\sum_j(w_j/W)(c_{n+1}/c_j)y_j$ plus the complementary weight times $c_{n+1}\alpha/\beta$. The data must lie on their count lattice; impossible noninteger recovered counts have zero <likelihood>.
// Destination: probability-theory.bigb
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