Unequal-Exposure Bayesian Credibility and Buhlmann-Straub

Why Unequal Exposure Changes the Average

A risk observed for several periods may have different amounts of exposure in each period. For example, a hypothetical policy may be in force for only part of one year, but for a full year in another. A period with more exposure contains more information about the risk’s claim rate, so the periods should not generally receive equal weights. In a repeated Poisson model, the natural weight is exposure.

This tutorial develops that result from a Gamma prior and connects it to the unequal-exposure Buhlmann-Straub credibility formula. The setting is deliberately specific: one risk has a constant claim rate across the observation periods, while the known exposure varies.

Model and assumptions: Let \(\theta\) be a risk’s claim frequency per unit of exposure. Conditional on \(\theta\), the observed counts \(N_1,\ldots,N_k\) are independent, with \(N_j\mid\theta\sim\operatorname{Poisson}(w_j\theta)\), where \(w_j>0\) is known exposure. Assume \(\theta\sim\operatorname{Gamma}(\alpha,\beta)\) in the shape-rate parameterization, with density proportional to \(\theta^{\alpha-1}e^{-\beta\theta}\), \(\alpha>0\), and \(\beta>0\). Thus \(E[\theta]=\alpha/\beta\) and \(\operatorname{Var}(\theta)=\alpha/\beta^2\).

The rate parameter \(\beta\) has units of exposure, while \(\alpha\) is dimensionless. This parameterization matters: the posterior rate adds observed exposure to \(\beta\). Define total claims \(N=\sum_{j=1}^k N_j\), total exposure \(W=\sum_{j=1}^k w_j\), and the observed exposure-weighted rate \(\bar{X}_w=N/W\).

Posterior Updating Across Unequal Exposures

Conditional independence gives a likelihood proportional to \(\theta^N e^{-W\theta}\). Multiplying by the Gamma prior kernel gives a Gamma posterior. Both the total number of claims and total exposure enter, rather than the separate period rates as an unweighted list.

\begin{aligned} L(\theta\mid N_1,\ldots,N_k) &\propto \prod_{j=1}^k \theta^{N_j}e^{-w_j\theta} =\theta^N e^{-W\theta},\\ \pi(\theta\mid N_1,\ldots,N_k) &\propto \theta^{\alpha+N-1}e^{-(\beta+W)\theta},\\ \theta\mid N_1,\ldots,N_k &\sim \operatorname{Gamma}(\alpha+N,\beta+W). \end{aligned}

The posterior mean is therefore \((\alpha+N)/(\beta+W)\). To reveal its credibility structure, write the prior mean as \(m=\alpha/\beta\), and use \(N=W\bar{X}_w\):

E[\theta\mid N_1,\ldots,N_k] =\frac{\alpha+N}{\beta+W} =\frac{\beta}{\beta+W}m+\frac{W}{\beta+W}\bar{X}_w.

Key result: The posterior mean is a credibility-weighted average of the prior mean \(m\) and the observed exposure-weighted rate \(\bar{X}_w\). Its credibility factor is \(Z=W/(W+\beta)\), so the posterior mean is \((1-Z)m+Z\bar{X}_w\). The prior rate parameter \(\beta\) acts as prior exposure in this formula.

Why the Credibility Form Holds

The result follows from the conjugate update and a simple rearrangement. The likelihood contributes \(N\) claims and \(W\) exposure to the prior’s shape and rate parameters, respectively. Consequently, the updated mean combines prior claims \(\alpha\) and observed claims \(N\), divided by prior exposure \(\beta\) plus observed exposure \(W\). Since \(\alpha=\beta m\) and \(N=W\bar{X}_w\), the numerator is \(\beta m+W\bar{X}_w\). Dividing each term by \(\beta+W\) gives the two weights shown above. They are nonnegative and sum to one.

Combine the observations.
Sum the counts to obtain \(N\), and sum the exposures to obtain \(W\). The observed rate is \(N/W\).

Update the prior.
The posterior is \(\operatorname{Gamma}(\alpha+N,\beta+W)\), so its mean is \((\alpha+N)/(\beta+W)\).

Identify credibility.
Match the posterior mean to \((1-Z)m+Z\bar{X}_w\). This gives \(Z=W/(W+\beta)\).

Connection to Buhlmann-Straub Credibility

The Buhlmann-Straub model expresses each period’s experience as a rate \(X_j=N_j/w_j\). In this Poisson setting, conditional on \(\theta\), that rate has mean \(\theta\) and variance \(\theta/w_j\). Therefore, larger exposure reduces the conditional variance of the observed rate.

E[X_j\mid\theta]=\theta,\qquad \operatorname{Var}(X_j\mid\theta)=\frac{\theta}{w_j}.

In the Buhlmann-Straub form, the conditional variance is \(v(\theta)/w_j\). Here \(v(\theta)=\theta\), so the expected process variance is \(\operatorname{EPV}=E[v(\theta)]=m\). The variance of the hypothetical means is \(\operatorname{VHM}=\operatorname{Var}(\theta)\). The associated credibility constant is \(K=\operatorname{EPV}/\operatorname{VHM}\), and the credibility factor for total exposure \(W\) is \(W/(W+K)\).

\operatorname{EPV}=m=\frac{\alpha}{\beta},\qquad \operatorname{VHM}=\frac{\alpha}{\beta^2},\qquad K=\frac{\operatorname{EPV}}{\operatorname{VHM}}=\beta.

Thus the Buhlmann-Straub factor \(W/(W+K)\) equals the Bayesian factor \(W/(W+\beta)\). Its weighted average of the period rates is \(\sum_j w_jX_j/W=\sum_jN_j/W=N/W\), exactly the observed rate in the posterior-mean expression. This match depends on the specified Poisson-Gamma model; it is not a claim that all Bayesian models reduce to this Buhlmann-Straub form.

Worked Example: Two Periods with Different Exposure

A hypothetical risk has claim rate \(\theta\) per exposure unit. Its prior is \(\operatorname{Gamma}(8,4)\), with shape 8 and rate 4. The risk produces 1 claim over exposure 1 and 9 claims over exposure 3. Find the posterior mean rate and express it as a credibility-weighted average. Assume conditionally independent Poisson counts and round the final rate to four decimal places.

Method. Pool claims and exposure, update the Gamma parameters, and then verify the credibility representation.

The prior mean is \(m=8/4=2\). Total claims are \(N=1+9=10\), and total exposure is \(W=1+3=4\). Hence the observed rate is \(\bar{X}_w=10/4=2.5\).

The posterior is \(\operatorname{Gamma}(8+10,4+4)=\operatorname{Gamma}(18,8)\). Its mean is \(18/8=2.25\).

The credibility factor is \(Z=4/(4+4)=0.5\). The credibility expression gives \((1-0.5)(2)+0.5(2.5)=1+1.25=2.25\). The posterior mean rate is 2.2500 claims per exposure unit.

Check. The posterior calculation and the weighted-average calculation agree. The common mistake to avoid is giving the two periods equal weight even though their exposures differ.

Interpreting the Result and Its Limits

The posterior mean is a premium rate, not a predicted count for an arbitrary future period. For a future period with known exposure \(w_*\), conditional on the observed data, the posterior predictive mean count is \(w_*E[\theta\mid N_1,\ldots,N_k]\). This follows because the future count has conditional mean \(w_*\theta\); taking its posterior expectation multiplies the posterior mean rate by \(w_*\).

The formula assumes the same latent rate \(\theta\) applies throughout the periods, exposures are known, and counts are conditionally independent Poisson variables. If the rate changes by period, exposure is measured incorrectly, or counts have additional conditional variation, this update need not be appropriate. A different model may be needed rather than an adjustment to the weights.

Worked Example: Posterior Rate and a Future Expected Count

A hypothetical equipment portfolio has a prior claim rate \(\theta\sim\operatorname{Gamma}(3,1.5)\), with shape 3 and rate 1.5. It records 0 claims over exposure 0.5, 4 claims over exposure 1.5, and 9 claims over exposure 3. Find the posterior mean rate and the expected count over a future exposure of 2. Assume conditionally independent Poisson counts. Round the rate and expected count to four decimal places.

Method. Use total claims and exposure for the posterior rate, then multiply by future exposure to obtain the predictive mean count.

The prior mean is \(3/1.5=2\). Total claims are \(N=0+4+9=13\); total observed exposure is \(W=0.5+1.5+3=5\). Thus the observed rate is \(13/5=2.6\).

The posterior distribution is \(\operatorname{Gamma}(3+13,1.5+5)=\operatorname{Gamma}(16,6.5)\). The posterior mean rate is \(16/6.5=32/13=2.461538\ldots\).

The credibility factor is \(Z=5/(5+1.5)=10/13\). The weighted-average check is \(2+(10/13)(2.6-2)=2+(10/13)(0.6)=2+6/13=32/13\), as required.

For future exposure \(w_*=2\), the expected count is \(2(32/13)=64/13=4.923076\ldots\). The answers are 2.4615 claims per exposure unit and 4.9231 expected claims. The second value is a mean, so it need not be an integer.

Check. The future exposure scales the rate; it does not change the posterior credibility factor based on the already observed data. Confusing the future exposure with observed exposure would incorrectly alter the posterior update.

Worked Example: Recovering the Buhlmann-Straub Parameters

For a hypothetical risk, \(\theta\sim\operatorname{Gamma}(6,3)\), with shape 6 and rate 3. The risk has 3 claims over exposure 2 and 4 claims over exposure 1. Calculate the Buhlmann-Straub EPV, VHM, credibility constant, and credibility premium rate. Then verify the result by Bayesian updating. Assume conditionally independent Poisson counts and round the final premium rate to four decimal places.

Method. First calculate the structural quantities from the Gamma prior and Poisson conditional variance. Then compare the Buhlmann-Straub credibility estimate with the posterior mean.

The prior mean is \(m=6/3=2\), and the prior variance is \(6/3^2=6/9=2/3\). Since the conditional variance of a period rate is \(\theta/w_j\), the EPV is \(E[\theta]=2\), while the VHM is \(\operatorname{Var}(\theta)=2/3\). Therefore \(K=\operatorname{EPV}/\operatorname{VHM}=2/(2/3)=3\).

Total exposure is \(W=2+1=3\), and the exposure-weighted observed rate is \((3+4)/3=7/3\). The credibility factor is \(Z=3/(3+3)=1/2\). The Buhlmann-Straub estimate is \((1-1/2)(2)+(1/2)(7/3)=1+7/6=13/6\).

For the Bayesian check, the posterior is \(\operatorname{Gamma}(6+7,3+3)=\operatorname{Gamma}(13,6)\), whose mean is \(13/6\). Both methods give \(13/6=2.166666\ldots\), or 2.1667 claims per exposure unit.

Check. The constant \(K=3\) equals the prior rate parameter \(\beta\). Using a reciprocal parameterization without converting it would produce an incorrect credibility factor.

Common Exam Trap: Averaging Period Rates Equally

Suppose one period has exposure 1 and 2 claims, while another has exposure 3 and 3 claims. The period rates are 2 and 1, but their unweighted average is \(1.5\). The correct pooled rate is \((2+3)/(1+3)=5/4=1.25\), because the second period represents three times as much exposure.

Exam trap: Do not replace \(\sum_jN_j/\sum_jw_j\) with the arithmetic mean of \(N_j/w_j\). The correct rate gives each period weight proportional to its exposure. Under the Gamma-Poisson model, this is the rate that enters the posterior mean.

Check Your Understanding

For each question, use the shape-rate Gamma parameterization and assume conditionally independent Poisson counts with known exposures.

A prior is \(\operatorname{Gamma}(4,2)\). A risk has 2 claims over exposure 1 and 4 claims over exposure 2. Find the posterior mean rate and credibility factor. (Answer: posterior mean \(=2.0000\); \(Z=0.6000\).)
A prior is \(\operatorname{Gamma}(2,1)\). A risk has 1 claim over exposure \(0.5\), 3 over exposure \(1.5\), and 5 over exposure 2. Find its posterior mean rate using the observed rate and credibility form. (Answer: \(2.2000\) claims per exposure unit.)
For a Gamma-Poisson model with prior \(\operatorname{Gamma}(5,2)\), find the EPV, VHM, and Buhlmann-Straub credibility constant. (Answer: EPV \(=2.5\), VHM \(=1.25\), \(K=2\).)
A prior is \(\operatorname{Gamma}(3,2)\). A risk has 2 claims over exposure 1 and 3 claims over exposure 3. Calculate the posterior mean rate and the credibility-weighted experience rate. (Answer: posterior mean \(=1.3333\) claims per exposure unit.)
A prior is \(\operatorname{Gamma}(4,2)\). A risk has 5 claims over total observed exposure 3. Find the posterior mean rate and the expected count over a future exposure of 1.5. (Answer: posterior mean rate \(=1.8000\); expected future count \(=2.7000\).)

What You'll Learn