Why Elicit Prior Parameters?
Bayesian credibility calculations often specify a prior distribution by its parameters, such as \(a\) and \(b\) for a beta prior. In practice, an actuary may find it easier to state a prior mean and how much weight the prior should carry. Elicitation translates those judgments into distribution parameters and makes the implied uncertainty visible.
The word strength needs care. For a beta prior on a probability, strength conventionally means the sum of the two shape parameters. For a gamma prior on a claim rate, the shape parameter acts as prior pseudo-claims, while the rate parameter acts as prior pseudo-exposure. These quantities are not interchangeable: the gamma parameters have units and depend on the rate’s exposure scale.
Eliciting a Beta Prior
A beta prior is useful for an unknown probability, such as the probability that a hypothetical insured event occurs in one policy period. Given a prior mean \(m\) and strength \(s\), the parameters follow from two equations: \(a+b=s\) and \(a/(a+b)=m\). Thus \(a\) is the mean multiplied by strength, and \(b\) is the remaining strength.
The conditions are \(0<m<1\) and \(s>0\), which ensure \(a>0\) and \(b>0\). A larger \(s\), with \(m\) fixed, concentrates the prior more tightly around its mean. It does not change the prior mean.
One way to verify the variance is to use the beta moments \(E[p]=a/(a+b)\) and \(E[p^2]=a(a+1)/((a+b)(a+b+1))\). Subtracting the square of the mean gives
Substituting \(a=ms\), \(b=(1-m)s\), and \(a+b=s\) yields \(m(1-m)/(s+1)\). The prior’s mean and strength therefore determine both its parameters and its variance; the variance is not a separate free choice once those inputs are fixed.
Worked Example: Eliciting a Beta Prior for an Event Probability
An actuary considers a hypothetical claim event with prior mean probability \(m=0.25\). The requested beta strength is \(s=20\). Find the beta parameters, prior variance, and equivalent pseudo-trials. State the method and avoid the common variance-denominator trap.
Method. Use \(a=ms\) and \(b=(1-m)s\):
The parameters are positive and sum to \(5+15=20\), as required. The mean checks:
The variance is
To six decimal places, the variance is \(0.008929\). The beta strength corresponds to 20 pseudo-trials: 5 pseudo-successes and 15 pseudo-failures, since those counts sum to the shape parameters. This interpretation is a useful way to understand the prior’s weight, not a claim that actual observations were collected.
Check and trap avoided. The recovered mean and strength match the inputs. Dividing \(0.1875\) by \(20\) would give \(0.009375\), but that incorrectly omits the \(+1\) in the beta variance formula.
Eliciting a Gamma Prior for a Claim Rate
For exponential claim severities, a gamma prior is often placed on the exponential rate \(\lambda\). The parameterization must be stated: here \(\beta\) is the rate parameter of the gamma distribution, not its scale. If a hypothetical severity \(X\) has exponential density \(f_X(x\mid\lambda)=\lambda e^{-\lambda x}\) for \(x\geq 0\), then \(\lambda\) has units inverse to those of \(X\).
Suppose the prior mean rate is \(m_\lambda\), and the stated gamma strength \(\alpha\) is the number of pseudo-claims. Since \(E[\lambda]=\alpha/\beta=m_\lambda\), the required gamma rate parameter is \(\beta=\alpha/m_\lambda\). Equivalently, if the prior mean and \(\beta\) are supplied, the pseudo-claim strength is \(\alpha=m_\lambda\beta\).
The connection with pseudo-exposure follows from the likelihood. For \(n\) independent exponential observations \(x_1,\ldots,x_n\), the likelihood as a function of \(\lambda\) is proportional to \(\lambda^n e^{-\lambda\sum_i x_i}\). Multiplication by the gamma prior density contributes \(\lambda^{\alpha-1}e^{-\beta\lambda}\). The posterior kernel therefore contains the combined terms \(\lambda^{\alpha+n-1}e^{-(\beta+\sum_i x_i)\lambda}\). The prior contributes \(\alpha\) to the claim-count term and \(\beta\) to the total-exposure term. Accordingly, \(\beta\) is pseudo-exposure, measured in the same units as each \(x_i\), and \(\alpha/\beta\) is the prior rate.
This is why a gamma prior’s strength cannot be reported without clarifying its meaning and the rate units. In this tutorial, gamma strength means \(\alpha\), the pseudo-claim count. In some settings, an actuary may instead be given a pseudo-exposure \(\beta\); then the corresponding shape must be calculated from the mean and that exposure.
Worked Example: Gamma Rate Prior with a Stated Pseudo-Claim Strength
For a hypothetical exponential-severity model, express claim amounts in thousands of dollars. The prior mean rate is \(m_\lambda=0.04\) per thousand dollars, and the prior strength is \(\alpha=12\) pseudo-claims. Find the gamma rate parameter, prior variance, and equivalent pseudo-exposure. Use the shape-rate parameterization.
Method. Match the prior mean \(\alpha/\beta\) to \(0.04\):
Thus \(\lambda\sim\operatorname{Gamma}(12,300)\), where \(\lambda\) is measured per thousand dollars and \(\beta\) is measured in thousand dollars. The implied variance is
Equivalently, \(m_\lambda^2/\alpha=0.04^2/12=0.0016/12=0.000133333\ldots\), which checks the variance calculation. The prior represents 12 pseudo-claims over 300 thousand dollars of pseudo-exposure. Their ratio is \(12/300=0.04\) claims per thousand dollars, matching the stated mean rate.
Check and trap avoided. Both parameter units and the mean are consistent. Treating \(\beta=300\) as a scale would reverse the mean formula and give an incompatible result; the calculation uses the specified shape-rate convention.
Recovering Strength from a Variability Judgment
Sometimes the prior mean is easier to judge than the strength. A coefficient of variation can help express the desired relative uncertainty. For a positive random variable, the coefficient of variation is its standard deviation divided by its mean. Under the gamma shape-rate parameterization, the variance formula implies
Consequently, a stated gamma coefficient of variation determines the shape strength as \(\alpha=1/\operatorname{CV}(\lambda)^2\). Once \(\alpha\) and the prior mean rate are known, calculate \(\beta=\alpha/m_\lambda\). This conversion applies to the gamma prior on the rate; it should not be confused with a coefficient of variation for the severity itself.
Worked Example: Eliciting Gamma Strength from Relative Uncertainty
A hypothetical portfolio uses an exponential claim-severity model measured in thousands of dollars. The prior mean rate is \(0.025\) per thousand dollars, and the actuary wants the gamma prior’s coefficient of variation to be \(0.5\). Find its shape and rate parameters, variance, and pseudo-exposure. State which uncertainty judgment is being used.
Method. The judgment is about relative uncertainty in \(\lambda\), not in individual claim amounts. Since \(\operatorname{CV}(\lambda)=1/\sqrt{\alpha}\),
Match the mean \(\alpha/\beta=0.025\):
The prior is \(\operatorname{Gamma}(4,160)\), with variance
The standard deviation is \(\sqrt{0.00015625}=0.0125\), so the coefficient of variation is \(0.0125/0.025=0.5\), as required. Its equivalent prior information is 4 pseudo-claims over 160 thousand dollars of pseudo-exposure, giving \(4/160=0.025\) per thousand dollars. Thus the parameters satisfy both the mean and relative-uncertainty judgments.
Check and trap avoided. The recalculated mean and coefficient of variation both match the inputs. Using \(\alpha=\operatorname{CV}^2\) instead of its reciprocal would yield \(0.25\), which is not the gamma shape implied by the variance formula.
Choosing and Checking a Strength
A stated strength should represent a defensible amount of prior information, not merely a convenient parameter value. After eliciting parameters, check the mean, strength, variance, and units. The following comparisons summarize the two conventions used here.
| Prior | Mean and strength inputs | Parameters | Implied variance |
|---|---|---|---|
| \(\operatorname{Beta}(a,b)\) | Mean \(m\), strength \(s\) | \(a=ms,\ b=(1-m)s\) | \(m(1-m)/(s+1)\) |
| \(\operatorname{Gamma}(\alpha,\beta)\), shape-rate | Mean \(m_\lambda\), pseudo-claims \(\alpha\) | \(\beta=\alpha/m_\lambda\) | \(m_\lambda^2/\alpha\) |
A beta strength counts pseudo-trials, divided into pseudo-successes and pseudo-failures. A gamma shape counts pseudo-claims, while its rate parameter represents pseudo-exposure. In the gamma case, the strength alone does not specify pseudo-exposure; the prior mean is needed to obtain \(\beta\). If a prompt instead specifies pseudo-exposure, use \(\alpha=m_\lambda\beta\).
A useful final check is to reconstruct each input from the parameters. For a beta prior, verify \(a+b=s\) and \(a/(a+b)=m\). For a gamma rate prior, verify \(\alpha/\beta=m_\lambda\), check the units of \(\beta\), and calculate the variance from the chosen parameterization. These checks catch both arithmetic errors and mismatches between a rate and a scale.
Check Your Understanding
For each hypothetical setting, identify the relevant prior parameterization and verify the requested implication.
- A beta prior for a claim probability has mean \(0.40\) and strength \(25\). Find \(a\), \(b\), and the prior variance. (Answer: \(a=10,\ b=15,\ \operatorname{Var}(p)=0.009231\) to six decimal places.)
- A beta prior has mean \(0.12\) and strength \(49\). Find its two shape parameters and variance. (Answer: \(a=5.88,\ b=43.12,\ \operatorname{Var}(p)=0.002112\).)
- A gamma shape-rate prior for a hypothetical claim rate has mean \(0.03\) per thousand dollars and shape strength \(\alpha=9\). Find \(\beta\), the variance of the rate, and pseudo-exposure. (Answer: \(\beta=300\) thousand dollars, \(\operatorname{Var}(\lambda)=0.0001\), and pseudo-exposure \(=300\) thousand dollars.)
- A gamma prior on a hypothetical rate measured per thousand dollars has mean \(0.08\) and coefficient of variation \(0.25\). Find \(\alpha\), \(\beta\), and the prior variance. (Answer: \(\alpha=16,\ \beta=200\) thousand dollars, and \(\operatorname{Var}(\lambda)=0.0004\).)
- A beta prior has \(a=7\) and \(b=13\). Find its mean, strength, and variance, and state how many pseudo-successes and pseudo-failures its parameters represent. (Answer: mean \(=0.35\), strength \(=20\), variance \(=0.35(0.65)/21=0.010833\) to six decimal places, with 7 pseudo-successes and 13 pseudo-failures.)