Bayesian Credibility

Study guide: Exam MAS-II

Bayesian Credibility applies Bayesian inference to credibility: the prior distribution of the risk parameter $\theta$ is updated with observed data to produce a posterior distribution, and the credibility estimate is the posterior mean $E[\theta \mid \mathbf{X}]$ .

$\text{Credibility Estimate} = E[\theta \mid X_1, \ldots, X_n]$

$\pi(\theta \mid \mathbf{X}) \propto L(\mathbf{X} \mid \theta) \cdot \pi(\theta)$

When the prior and likelihood form a conjugate pair, the posterior mean is a linear credibility formula of the form $Z\bar{X} + (1-Z)\mu_0$
In this case, Bayesian credibility = Bühlmann Credibility (Bayesian is optimal; Bühlmann is the best linear approximation, and they agree for conjugate priors)

Common conjugate pairs:

Likelihood	Prior	Posterior
Poisson $(\theta)$	Gamma $(\alpha, \beta)$	Gamma $(\alpha + \sum x_i,\, \beta + n)$
Binomial $(m, \theta)$	Beta $(\alpha, \beta)$	Beta $(\alpha + \sum x_i,\, \beta + mn - \sum x_i)$
Normal $(\theta, \sigma^2)$	Normal $(\mu_0, \tau^2)$	Normal (linear combination)

Bayesian Update for Poisson Claim Frequency A policyholder's annual claim count $N \sim \text{Poi}(\theta)$ . The prior for $\theta$ is Gamma $(2, 0.5)$ (mean $= 2 \times 0.5 = 1$ ). In 3 years, the policyholder had 0, 2, and 1 claims. Find the posterior mean.

Answer $\sum x_i = 3$ , $n = 3$ . The prior's $0.5$ is a scale, so its rate is $\beta = 1/0.5 = 2$ . The Poisson–gamma update adds the claim count to the shape and the years to the rate: the posterior is Gamma with shape $2 + 3 = 5$ and rate $2 + 3 = 5$ (scale $0.2$ ). Posterior mean $= 5/5 = 1$ claim per year. This is the credibility-weighted blend: posterior mean $= Z \cdot \bar{X} + (1-Z)\mu_0$ with $Z = n/(n + \beta) = 3/(3 + 2) = 0.6$ , giving $0.6(1) + 0.4(1) = 1$ . Here the sample mean equals the prior mean, so the data confirm the prior rather than move it.