Inflation-adjusted Poisson-gamma credibility (${\hat{Y}}_{n+1}=c_{n+1}\frac{\alpha +{\sum }_j{w}_j{y}_j/c_j}{\beta +{\sum }_j{w}_j}$)

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 $k_j=w_j{y}_j/c_j$. The posterior distribution shape is $\alpha +{\sum }_j{k}_j$ and rate is $\beta +{\sum }_j{w}_j$. 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