Bayesian Credibility Calculation Workshop

From Model Inputs to a Risk-Level Estimate

Bayesian credibility calculations combine two ingredients: prior information about a risk and experience observed for that risk. The calculation is complete only when the model, units, prior parameters, data, posterior distribution, and resulting risk-level estimate are all clear. This workshop develops that sequence for hypothetical claim-count and claim-incidence examples.

The risk-level estimate in these examples is the posterior mean of the risk’s unknown claim rate or claim probability. It is a credibility-weighted blend of the prior mean and the observed experience rate. It is not, by itself, a distribution for future claim outcomes.

Definition: In the Gamma-Poisson model, a risk has an unknown claim rate \(\lambda\), with \(\lambda\sim\operatorname{Gamma}(\alpha,\beta)\) in the shape-rate parameterization. Conditional on \(\lambda\), the claim count \(N\) over exposure \(t\) has a Poisson distribution with mean \(t\lambda\). In the Beta-Binomial model, a risk has an unknown claim probability \(p\), with \(p\sim\operatorname{Beta}(\alpha,\beta)\). Conditional on \(p\), the number \(S\) of claim occurrences in \(m\) independent trials has a Binomial distribution with parameters \(m\) and \(p\).

For a Gamma distribution written with shape \(\alpha\) and rate \(\beta\), the mean is \(\alpha/\beta\), the variance is \(\alpha/\beta^2\), and the density is proportional to \(\lambda^{\alpha-1}e^{-\beta\lambda}\). The rate parameter has units reciprocal to the exposure unit used for \(\lambda\). In particular, the posterior rate adds exposure, not claim count.

For a Beta distribution, the mean is \(\alpha/(\alpha+\beta)\), and the variance is \(\alpha\beta/[(\alpha+\beta)^2(\alpha+\beta+1)]\). The sum \(\alpha+\beta\) is often used as a prior strength, measured in equivalent Bernoulli trials. These interpretations help translate actuarial information into parameters; they do not mean that the prior represents literal observed claims or trials.

Gamma-Poisson Updating

Assume the risk’s claim count, conditional on its rate \(\lambda\), is Poisson over exposure \(t\), and the prior for \(\lambda\) is Gamma with shape \(\alpha\) and rate \(\beta\). For observed count \(n\), the posterior distribution is Gamma with shape \(\alpha+n\) and rate \(\beta+t\). Its posterior mean is the risk-level estimate.

\lambda\mid N=n \sim \operatorname{Gamma}(\alpha+n,\beta+t), \qquad E[\lambda\mid N=n]=\frac{\alpha+n}{\beta+t}.

To express this mean as credibility, define the prior mean \(\mu=\alpha/\beta\), the observed rate \(\bar{x}=n/t\) when \(t>0\), and the credibility weight \(Z=t/(\beta+t)\). Substitution gives

\frac{\alpha+n}{\beta+t} = \frac{\beta}{\beta+t}\frac{\alpha}{\beta} + \frac{t}{\beta+t}\frac{n}{t} = (1-Z)\mu+Z\bar{x}.

Thus, more exposure increases the weight on the observed rate, while a larger prior rate parameter \(\beta\) gives the prior more equivalent exposure and reduces the weight on current experience. This formula assumes \(t>0\) for the observed rate representation; the posterior formula itself also applies at \(t=0\), in which case the posterior equals the prior.

Set the exposure units.
For example, if \(\lambda\) is claims per vehicle-year, express \(t\) in vehicle-years.

Specify the prior.
Record \(\alpha\) and the shape-rate parameter \(\beta\), or convert a prior mean and strength into those parameters.

Update and estimate.
Add the count to the shape and exposure to the rate; compute the posterior mean and, as a check, its credibility-weighted form.

Worked Example: Updating a Vehicle Claim Rate

A hypothetical portfolio segment has a prior mean claim rate of \(0.08\) claims per vehicle-year and a prior strength of 4 vehicle-years. Assume its unknown rate \(\lambda\) has a Gamma shape-rate prior, and conditional claim counts follow a Poisson model. The segment then produces 9 claims over 100 vehicle-years. Find the prior parameters, posterior distribution, credibility weight, posterior mean, and posterior variance.

Prior parameters. For a Gamma shape-rate prior, the prior mean is \(\alpha/\beta\), and the stated strength is \(\beta\). Therefore,

\beta=4,\qquad \alpha=0.08(4)=0.32.

So \(\lambda\sim\operatorname{Gamma}(0.32,4)\). The observed rate is \(9/100=0.09\) claims per vehicle-year.

Posterior distribution. Add the observed claim count to the shape and the exposure to the rate:

\alpha_{\mathrm{post}}=0.32+9=9.32,\qquad \beta_{\mathrm{post}}=4+100=104.

Thus \(\lambda\mid N=9\sim\operatorname{Gamma}(9.32,104)\), and its posterior mean is \(9.32/104=0.0896154\) claims per vehicle-year.

Credibility check. The weight on the observed rate is \(Z=100/(4+100)=100/104=0.961538\). The weighted estimate is

$$ (1-0.961538)(0.08)+(0.961538)(0.09) =0.0030769+0.0865384 =0.0896153. $$

The tiny difference from the direct calculation is due to displayed intermediate rounding. The posterior variance is \(9.32/104^2=9.32/10{,}816\approx0.0008617\) (rounded to seven decimal places). The direct and credibility-weighted posterior means agree before rounding. The method is Gamma-Poisson conjugate updating; a common mistake it avoids is treating the raw observed rate \(0.09\) as the estimate without allowing for prior information.

Beta-Binomial Updating

Assume each trial is an independent Bernoulli outcome conditional on the risk’s claim probability \(p\), with the same probability \(p\) for each trial. With a Beta prior and \(s\) observed claim occurrences in \(m\) trials, the posterior is Beta with parameters \(\alpha+s\) and \(\beta+m-s\). The posterior mean is a credibility-weighted combination of the prior mean and observed proportion \(s/m\).

p\mid S=s \sim \operatorname{Beta}(\alpha+s,\beta+m-s), \qquad E[p\mid S=s]=\frac{\alpha+s}{\alpha+\beta+m}.

Write the prior mean as \(\mu=\alpha/(\alpha+\beta)\), the prior strength as \(k=\alpha+\beta\), and \(Z=m/(k+m)\). Then

\frac{\alpha+s}{k+m} = \frac{k}{k+m}\frac{\alpha}{k} + \frac{m}{k+m}\frac{s}{m} = (1-Z)\mu+Z\frac{s}{m}.

The result requires \(m>0\) to define the observed proportion. The posterior formula remains valid when \(m=0\), returning the prior mean. The trials must represent comparable opportunities with a common underlying probability; a model with different probabilities across trials needs a different likelihood.

Worked Example: Estimating a Claim Probability for a Class of Risks

A hypothetical insurer uses a prior mean claim probability of \(0.12\) for a class of independently observed risks, with prior strength 40 trials. Suppose the class is modeled using a common probability \(p\), a Beta prior, and a Binomial likelihood. Among 50 observed trials, 8 have claims. Calculate the prior parameters, posterior distribution, credibility weight, posterior mean, and posterior variance.

Prior parameters. The strength is \(\alpha+\beta=40\), and the mean is \(\alpha/40=0.12\). Hence \(\alpha=4.8\) and \(\beta=40-4.8=35.2\). The prior is \(\operatorname{Beta}(4.8,35.2)\).

Posterior distribution. There are 8 claim occurrences and \(50-8=42\) non-occurrences. Therefore,

p\mid S=8\sim\operatorname{Beta}(4.8+8,35.2+42) =\operatorname{Beta}(12.8,77.2).

The posterior mean is \(12.8/(12.8+77.2)=12.8/90=0.142222\). The observed proportion is \(8/50=0.16\), and the credibility weight is \(Z=50/(40+50)=5/9\). The weighted calculation checks the result:

\left(\frac{4}{9}\right)(0.12)+\left(\frac{5}{9}\right)(0.16) =0.053333+0.088889 =0.142222.

The posterior variance is \(12.8(77.2)/[90^2(91)]=988.16/737{,}100\approx0.001341\) (rounded to six decimal places). The calculation uses the Beta-Binomial model and treats each trial as one comparable opportunity. A mistake to avoid is adding all 50 trials to the Beta shape for claim occurrences: only the 8 successes are added to \(\alpha\); the 42 non-occurrences are added to \(\beta\).

Prior Strength and a Shape-Rate Trap

A prior mean and a prior strength are useful summaries, but their meanings depend on the model. In the Gamma-Poisson model, \(\beta\) has exposure units and controls the prior’s weight relative to observed exposure. In the Beta-Binomial model, \(\alpha+\beta\) is the prior strength in equivalent trials. These are not interchangeable parameterizations.

Worked Example: Sparse Count Experience with an Explicit Gamma Prior

For a hypothetical risk, suppose \(\lambda\), measured in claims per exposure unit, has prior distribution \(\operatorname{Gamma}(3,30)\) in shape-rate form. Assume its count over exposure \(t\), conditional on \(\lambda\), is Poisson. The risk has 1 claim over 20 exposure units. Find the posterior distribution, credibility weight, and risk-level estimate, and identify the effect of confusing count with exposure in the update.

Prior mean and observed rate. The prior mean is \(3/30=0.10\) claims per exposure unit. The observed rate is \(1/20=0.05\).

Correct update. The claim count updates the shape, and exposure updates the rate:

\lambda\mid N=1\sim\operatorname{Gamma}(3+1,30+20) =\operatorname{Gamma}(4,50).

The posterior mean is \(4/50=0.08\). The credibility weight is \(Z=20/(30+20)=0.40\), so the weighted estimate is \((1-0.40)(0.10)+(0.40)(0.05)=0.060+0.020=0.08\). The posterior variance is \(4/50^2=4/2500=0.0016\).

Trap and correction. Adding the claim count to the prior rate would incorrectly give \(\operatorname{Gamma}(4,31)\), with mean \(4/31\approx0.1290\), which is above the prior mean despite an observed rate of \(0.05\). This is wrong because the likelihood contributes \(t\lambda\) in its exponential term, so the rate update is \(30+t=50\), not \(30+n\). The correct posterior mean \(0.08\) lies between the prior mean \(0.10\) and observed rate \(0.05\), as a credibility-weighted estimate should.

How to Check a Completed Calculation

A reliable calculation should pass several checks before it is used. First, make sure that the likelihood matches the observation: counts over exposure call for a Poisson model, while successes in a fixed number of comparable trials call for a Binomial model. Second, check that the updated parameters use the right data totals. Third, compare the posterior mean with its credibility-weighted form.

For Gamma-Poisson updating, the posterior shape increases by the claim count and the posterior rate increases by exposure.
For Beta-Binomial updating, the first parameter increases by successes and the second by non-occurrences.
When there is positive experience, the posterior mean lies between the prior mean and the observed rate or proportion.
When experience is zero, the posterior distribution is still well-defined; the data update the relevant parameter even though an observed rate or proportion may not be defined.

These calculations estimate an unknown risk parameter conditional on the chosen model and observed experience. A question about the distribution of future counts or future claim occurrences asks for a predictive distribution, which is a different calculation.

Key takeaway: State the prior and likelihood first, update the correct parameters using the correct units, and compute the posterior mean. Then verify that the mean equals the credibility-weighted blend of prior and observed experience.

Check Your Understanding

For each problem, identify the conjugate model and show the parameter updates before checking the final answer.

A hypothetical risk has a Gamma shape-rate prior with mean \(0.06\) claims per exposure-year and prior strength 25 exposure-years. It has 4 claims over 50 exposure-years. Find the posterior distribution, credibility weight, and risk-level estimate. (Answer: \(\operatorname{Gamma}(5.5,75)\); \(Z=2/3\); posterior mean \(5.5/75=0.07333\) claims per exposure-year.)
A hypothetical risk has prior \(\operatorname{Gamma}(2.4,30)\) in shape-rate form and experiences zero claims over 15 exposure units. Under a conditional Poisson model, find the posterior distribution, credibility weight, and estimate. (Answer: \(\operatorname{Gamma}(2.4,45)\); \(Z=15/45=1/3\); posterior mean \(2.4/45=0.05333\) claims per exposure unit.)
A class has a Beta prior with mean \(0.08\) and strength 50. In 40 independent trials, 6 have claims. Find the posterior distribution, credibility weight, and estimate. (Answer: prior \(\operatorname{Beta}(4,46)\), posterior \(\operatorname{Beta}(10,80)\); \(Z=40/90=4/9\); posterior mean \(10/90=0.1111\).)
A hypothetical risk class has prior \(\operatorname{Beta}(3,27)\). In 60 comparable trials, 12 have claims. Find the posterior distribution and risk-level estimate, and verify the estimate as a weighted blend. (Answer: posterior \(\operatorname{Beta}(15,75)\); \(Z=60/(30+60)=2/3\); estimate \(15/90=0.1667=(1/3)(0.10)+(2/3)(0.20)\).)
A hypothetical classification has prior mean claim probability \(0.04\) and prior strength 80 trials. It records 10 claim occurrences in 100 independent trials. Find the prior parameters, posterior distribution, credibility weight, and risk-level estimate. (Answer: prior \(\operatorname{Beta}(3.2,76.8)\); posterior \(\operatorname{Beta}(13.2,166.8)\); \(Z=100/180=5/9\); posterior mean \(13.2/180=0.07333\).)

What You'll Learn