Bühlmann model ($m=\mathbb{E}[m(\Theta )],v=\mathbb{E}[v(\Theta )],a=\mathrm{Var}(m(\Theta ))$)

Given a latent risk parameter $\Theta$, observations are independent and identically distributed random variables with conditional expectation $m(\Theta )$ and conditional variance $v(\Theta )$. The Bühlmann credibility premium estimates $m(\Theta )$ by an affine function of past observations under mean squared error. Its population parameters are $m=\mathbb{E}m(\Theta )$, expected process variance $v=\mathbb{E}v(\Theta )$, and variance of hypothetical means $a=\mathrm{Var}(m(\Theta ))$.

Conditional on a common latent variable $\Theta$, annual averages $X_j$ are independent, have common conditional expectation $\mu (\Theta )$, and have conditional variance ${\sigma }^2(\Theta )/m_j$ for known positive exposures $m_j$. Its structural parameters are $m_0=\mathbb{E}\mu (\Theta )$, expected process variance $v=\mathbb{E}{\sigma }^2(\Theta )$ and variance of hypothetical means $a=\mathrm{Var}(\mu (\Theta ))$. It generalizes the equal-exposure Bühlmann model.

For total exposure $W={\sum }_j{m}_j$ and weighted average ${\overline{X}}_w={\sum }_j{m}_j{X}_j/W$, the best affine mean squared error estimate of the conditional mean is $Z{\overline{X}}_w+(1-Z)m_0$, where $Z=Wa/(Wa+v)$. If the coefficients sum to $B$, their error is $a(1-B{)}^2+v{\sum }_j{b}_j^2/m_j$. Cauchy-Schwarz inequality minimizes the latter term at $b_j=B{m}_j/W$, and a one-dimensional quadratic minimization then gives $B=Z$.

With a Poisson-gamma conjugacy with unequal exposures model, $m_0=v=\alpha /\beta$ and $a=\alpha /{\beta }^2$, so the Bühlmann–Straub credibility factor is $W/(W+\beta )$. The credibility estimate simplifies to $(\alpha +{\sum }_j{Y}_j)/(\beta +W)$, exactly the posterior mean and hence the Bayes estimator under squared error loss. The agreement is exact because that posterior mean is already affine in the exposure-weighted data.

The experience weight is $Z=Wa/(Wa+v)$, where $W$ is total observed exposure, $a$ the variance of hypothetical means, and $v$ the expected process variance. With positive $a$, it is $W/(W+v/a)$. More exposure raises the weight; larger process noise lowers it. If $a=0$, the unknown conditional mean is constant and one can set $Z=0$.

The Bühlmann credibility premium is the best affine mean squared error estimate of a risk’s conditional claim expected value. In the Bühlmann model it is $Z\overline{X}+(1-Z)m$, where $Z=na/(na+v)$. The linear least-squares projection equations use $\mathrm{Var}(X_j)=a+v$ and $\mathrm{Cov}(X_i,X_j)=a$ for distinct years.

For independent comparable risks observed over $n\ge 2$ years, estimate the expected process variance $v$ by the average within-risk sample variance. The sample variance $B$ of the risk means estimates $a+v/n$, where $a$ is the variance of hypothetical means. Thus $B-\hat{v}/n$ is unbiased for $a$; truncating at zero produces a practical nonnegative estimate but introduces bias. Use the grand mean for $m$ and plug these estimates into the Bühlmann credibility premium $\hat{Z}{\overline{X}}_s+(1-\hat{Z})\hat{m}$. At least two risks are needed for $B$; a single year cannot estimate within-risk variance this way.

Let a fixed policy intensity have prior density $3{\theta }^{-4}$ on $\theta >1$, and let $n$ annual claim counts be conditionally independent with Poisson distribution mean $\theta$. The expected process variance is $3/2$ and the variance of hypothetical means is $3/4$, so the Bühlmann credibility factor is $n/(n+2)$. A total count $k$ gives the displayed credibility premium. By contrast, the Bayesian posterior density is proportional to ${\theta }^{k-4}{e}^{-n\theta }$ on $\theta >1$. Its mean is the ratio ${\int }_1^{\infty }{\theta }^{k-3}{e}^{-n\theta }d\theta /{\int }_1^{\infty }{\theta }^{k-4}{e}^{-n\theta }d\theta$. For $n=2,k=5$, that Bayesian credibility estimate is $5/3$, whereas the affine estimate is $2$.

In the Gamma prior for an inverse-gamma scale model, assume $k>2$. The variance of hypothetical means is $a=\alpha /[{\lambda }^2(k-1{)}^2]$, and the expected process variance is $v=\alpha (\alpha +1)/[{\lambda }^2(k-1{)}^2(k-2)]$. The Bühlmann credibility factor is $Z=n/(n+v/a)$, giving the displayed expression. The optimal affine credibility estimate is $(1-Z)\alpha /[\lambda (k-1)]+Z\overline{X}$. Its fixed-weight arithmetic-mean form differs from the reciprocal-based posterior mean.

With fixed structural parameters and $K=v/a$, the Bühlmann credibility estimate has form $({\sum }_{i=1}^n{x}_i+K{m}_0)/(n+K)$. The next observation updates it by the displayed rule. Consequently the estimate decreases exactly when the new observation is less than the previous credibility estimate, with strict equality handled separately.

The credibility factor in the Bühlmann model is the weight $Z=na/(na+v)$ placed on the sample mean after $n$ observations. The remaining weight is placed on the population expected value. More experience or larger variance of hypothetical means increases $Z$; larger expected process variance decreases it.

The variance of hypothetical means in the Bühlmann model is the between-risk variance of the conditional claim expected value. It creates the shared covariance $a$ between different years of the same risk and determines how much individual experience should influence the premium.

The expected process variance in the Bühlmann model averages within-risk conditional variance over the population of risks. It measures variation that repeated observations can average out and contributes $v/n$ to the variance of the sample mean.