Explain and apply Bayesian (greatest accuracy) credibility.

Bayesian credibility is the gold standard. You compute the predictive mean given the data, full stop. Bühlmann is its linear shortcut. Knowing when each applies and which numbers feed the formula is the whole game.

The setup. A risk has parameter $\Theta$ drawn from a prior $\pi(\theta)$. Conditional on $\Theta = \theta$, claim counts or amounts $X_1, \dots, X_n$ are i.i.d. with conditional density $f_{X\mid\Theta}(x\mid\theta)$. You observe the data and want to predict $X_{n+1}$ for the same risk.

Hypothetical mean $\mu(\theta) = E[X\mid\Theta=\theta]$. This is the true mean for a risk whose parameter is $\theta$. It is unknown because $\theta$ is unknown.

Process variance $v(\theta) = \text{Var}(X\mid\Theta=\theta)$. This is the noise around the hypothetical mean for a fixed risk.

Three population summaries drive everything:

• EHM (Expected Hypothetical Mean): $\mu = E[\mu(\Theta)]$. The grand mean across the portfolio.
• EPV (Expected Process Variance): $v = E[v(\Theta)]$. Average within-risk noise.
• VHM (Variance of Hypothetical Means): $a = \text{Var}(\mu(\Theta))$. Between-risk dispersion.

KEY: EPV is "within," VHM is "between." If risks barely differ, VHM is small and credibility is low.

Common mistakes

• Computing $a$ as the sample variance of the data. The correct value is $\text{Var}(\mu(\Theta))$, built from the prior, not the data.
• Forgetting that $v$ is $E[v(\Theta)]$, not $\text{Var}(X)$. The unconditional variance is $v + a$, not $v$.
• Using $Z = n/(n+k)$ with Bühlmann-Straub when exposures vary. Use $Z = m/(m+k)$ with total exposure $m$.

Bottom line

• Bayesian premium is $E[X_{n+1} \mid \mathbf{X}]$, computed from the posterior of the risk parameter $\Theta$.
• Bühlmann is the least-squares (squared-error) linear approximation: $P_c = Z\bar{X} + (1-Z)\mu$ with $Z = n/(n+k)$, $k = v/a$.
• EPV $v = E[\text{Var}(X\mid\Theta)]$; VHM $a = \text{Var}(\mu(\Theta))$; EHM $\mu = E[\mu(\Theta)]$.
• All of $\mu$, $v$, $a$ come from the prior structure, never from the sample variance of the data.

Exam shortcut

For Poisson-Gamma$(\alpha, \theta)$: the Bayesian (and Bühlmann) premium after $n$ periods with total claims $\sum x_i$ is $(\alpha + \sum x_i)\theta/(1+n\theta)$. Memorize this; it appears constantly. When given a two-point prior, build a 4-column table: $\theta$, $\pi(\theta)$, $\mu(\theta)$, $v(\theta)$. Get $\mu$, $v$, $a$ in three lines. If the problem gives unconditional mean and variance and $v$, back out $a$ from $\text{Var}(X) = v + a$. No likelihood needed.

The full lesson (about 1,838 words, 12 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 5a

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