Bayesian Estimation and Credibility Theory Fundamentals

Bayesian Estimation in Insurance

In Bayesian credibility, the risk parameter Theta has a prior distribution pi(theta). Given Theta = theta, claims X1, ..., Xn are conditionally iid with density f(x|theta). The posterior distribution pi(theta|x) is proportional to L(x|theta) * pi(theta). The Bayesian estimate of the next period's expected claims is the posterior mean: E[X_{n+1}|X1, ..., Xn] = integral of mu(theta) * pi(theta|x) d_theta, where mu(theta) = E[X|Theta = theta].

For conjugate models, the posterior has the same form as the prior with updated parameters. This makes the Bayesian estimate easy to compute.

Connection to Credibility

In many conjugate models, the Bayesian estimate of the mean is a linear function of the sample mean: E[mu(Theta)|X1, ..., Xn] = Z * X_bar + (1 - Z) * mu_0, where Z is the credibility factor (0 <= Z <= 1), X_bar is the sample mean, and mu_0 is the prior mean. The credibility factor Z increases with sample size: more data means more weight on the observed experience. This linear credibility formula is exact for certain model combinations (e.g., Poisson-Gamma, Normal-Normal).

Key Quantities

The hypothetical mean is mu(theta) = E[X|Theta = theta]. The process variance is v(theta) = Var(X|Theta = theta). Expected value of the process variance: v = E[v(Theta)]. Variance of the hypothetical means: a = Var(mu(Theta)). For the exact credibility models on Exam STAM, the credibility factor Z = n / (n + v/a) = n / (n + k), where k = v/a. As n increases, Z approaches 1 (full credibility). Understanding these quantities and how they relate to the Bayesian posterior is essential for Exam STAM.