Bayesian credibility

Bayesian credibility estimates a risk's conditional mean by its Bayesian posterior expected value under squared-error loss. When that posterior mean is an affine combination of exposure-weighted individual experience and a prior mean, the coefficient of the experience is its credibility factor. Exact Bayesian credibility is distinct from merely selecting the best affine estimator when the true posterior mean is nonlinear.

Conditionally independent normal distribution observations of mean $\theta$ and known variance ${\sigma }_1^2$, with normal prior mean $\mu$ and variance ${\sigma }_2^2$, give normal-normal conjugacy with known observation variance. Their posterior mean has the displayed exact credibility estimate form. It minimizes posterior squared-error loss and predicts the conditional mean of the next observation.

For conditionally independent annual averages $X_j\mid \theta \sim N(\theta ,v/m_j)$ and prior $\theta \sim N(\mu ,{\sigma }^2)$, the posterior mean is $Z{\overline{X}}_w+(1-Z)\mu$, with ${\overline{X}}_w={\sum }_j{m}_j{X}_j/W$. Completing the quadratic likelihood gives posterior precision ${\sigma }^{-2}+W/v$. This is an exact Bayesian credibility version of the Bühlmann–Straub credibility estimate. Increasing the between-risk prior variance increases the experience weight; increasing process variance decreases it. The next total pure premium is $m_{n+1}$ times the estimated per-life mean.

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

For a common success chance with beta prior and conditionally independent binomial observations of sizes $m_i$, put $M={\sum }_i{m}_i$ and $s={\sum }_i{x}_i$. Beta-binomial conjugacy gives posterior mean $(\alpha +s)/(\alpha +\beta +M)$, equal to $Z(s/M)+(1-Z)\alpha /(\alpha +\beta )$. Multiplying by the next exposure predicts its expected count. The factor tends to one as exposure increases and to zero as prior concentration increases at fixed prior mean.