Choosing Prior Parameters From Actuarial Information

Why Prior Parameters Need Calibration

Actuarial information about heterogeneous risks is often expressed in practical terms: an average claim frequency, a range of plausible risk levels, a standard deviation, or a coefficient of variation. A Bayesian model, however, requires parameters for a probability distribution. Choosing those parameters is a moment-matching task: select a distributional family, translate the supplied assumptions into its parameters, and verify that the resulting distribution has the intended moments.

This tutorial considers two common choices. A Gamma prior can describe a positive, uncertain claim rate, while a Beta prior can describe an uncertain probability between zero and one. The focus is calibration of the prior distribution, not updating it with observed claims. The likelihood and posterior calculations are separate steps.

Gamma Priors for Claim Rates

Let \(\lambda\) be an unknown claim rate, such as expected claims per policy-year. Use a Gamma distribution with shape parameter \(\alpha>0\) and rate parameter \(\beta>0\), denoted \(\lambda\sim\operatorname{Gamma}(\alpha,\beta)\). In this convention, the density is proportional to \(\lambda^{\alpha-1}e^{-\beta\lambda}\), and the rate parameter \(\beta\), not its reciprocal, appears in the exponential.

Suppose actuarial information supplies a positive prior mean \(m\) and variance \(s^2\). Equating the Gamma moments to these assumptions gives

The equations have a unique positive solution when \(m>0\) and \(s^2>0\). A standard deviation can be used after squaring it to obtain the variance. If the assumptions instead give a coefficient of variation \(c=s/m\), then \(s^2=c^2m^2\), so the same formulas reduce to \(\alpha=1/c^2\) and \(\beta=1/(c^2m)\).

Why the Gamma Formulas Work

The moment-matching equations are obtained by solving the distribution's moment equations, rather than by treating shape and rate as informal labels. For a Gamma variable in the stated shape-rate convention, its mean is \(\alpha/\beta\) and its variance is \(\alpha/\beta^2\). The mean equation gives \(\alpha=m\beta\). Substitute this into the variance equation:

Then \(\alpha=m\beta=m^2/s^2\). This also explains the role of dispersion: at a fixed mean, a larger variance produces a smaller shape and a smaller rate. Both parameters change, but their ratio continues to preserve the assumed mean.

Units provide a useful independent check. If \(\lambda\) is measured in claims per policy-year, \(\beta\) has units of policy-years per claim-rate unit, so that \(\alpha/\beta\) has the units of \(\lambda\). Shape \(\alpha\) is dimensionless. If exposure is rescaled—for example, from policy-years to thousands of policy-years—the numerical rate parameter must be consistent with the new numerical unit for \(\lambda\).

Beta Priors for Claim Probabilities

Let \(p\) be an unknown claim probability, such as the probability that a hypothetical insured has at least one claim during a defined period. A Beta prior has parameters \(a>0\) and \(b>0\), written \(p\sim\operatorname{Beta}(a,b)\). Its support is \(0<p<1\), and its moments are

If the assumed mean is \(m\) and variance is \(s^2\), define the concentration \(k=a+b\). Since \(a=mk\) and \(b=(1-m)k\), substitution into the variance gives

For a non-degenerate Beta distribution, \(0<m<1\) and \(0<s^2<m(1-m)\). The upper bound follows because \(k\) must be positive; equality would require \(k=0\), which is not an allowed Beta parameter sum. If the assumed mean is at an endpoint, or the variance is zero, a point mass rather than an ordinary Beta distribution is needed. If the variance violates the bound, the assumptions cannot be represented by a Beta prior.

Feasibility Checks and Common Exam Traps

Parameter calculations are not complete until the resulting distribution is checked against the original actuarial assumptions. For Gamma calibration, verify that the mean and variance are positive and use a single parameter convention consistently. For Beta calibration, check the feasibility bound before calculating \(a\) and \(b\). These checks catch impossible inputs as well as arithmetic and interpretation errors.

• Confusing rate with scale. In the Gamma shape-rate convention used here, \(\beta=m/s^2\). The scale is \(s^2/m\), its reciprocal. Always substitute the proposed parameters into the stated moment equations.
• Using a standard deviation where a variance is required. Both Gamma and Beta formulas use \(s^2\). Square the supplied standard deviation first.
• Ignoring probability bounds. For a Beta prior, calculate \(m(1-m)\) and check that the assumed variance is strictly smaller. A mathematically infeasible pair of moments cannot be fixed by rounding parameters.
• Changing units without changing interpretation. A numerical claim rate per policy-year is not the same numerical quantity as claims per 1,000 policy-years. State the exposure unit alongside the rate and parameter values.
• Stopping after solving for parameters. Substitute them back into the distribution's mean and variance formulas. This final check is part of the calibration, not an optional presentation detail.

For example, a proposed Beta prior with mean \(0.08\) and standard deviation \(0.30\) is impossible: its variance is \(0.09\), whereas \(m(1-m)=0.08(0.92)=0.0736\), and a proper Beta distribution requires variance strictly below \(0.0736\). The problem is not a computational mistake; the requested moments are incompatible with the chosen family.