Bayesian updating and credibility

Bayes' theorem is the single mechanism by which evidence updates belief: $p(s\mid e) \propto L(e\mid s)\,p(s)$, where $L(e\mid s)=P(e\mid s)$ is the likelihood of the observed evidence under each state and $p(s)$ is the prior. Conjugate families make the arithmetic closed-form — a Beta prior with Binomial data gives a Beta posterior; a Gamma prior with Poisson claim counts gives a Gamma posterior; a Normal prior with Normal data gives a Normal posterior — and actuaries lean on these heavily in claims-count and severity modelling.

Credibility as linear Bayes. Full Bayesian updating requires a parametric likelihood and prior. Bühlmann's credibility theory (1967) asks a more modest question: among all linear estimators of the form $\hat{X}=Z\bar{X}+(1-Z)\mu$ (a weighted average of the observed experience $\bar X$ and the overall/manual mean $\mu$), which minimises mean squared error? The answer is the celebrated credibility factor

$Z = \frac{n}{n+k}, \qquad k = \frac{\text{Expected Value of Process Variance (EVPV)}}{\text{Variance of Hypothetical Means (VHM)}},$

which increases toward 1 as the volume of own experience $n$ grows relative to the noisiness of individual risks vs. the variability across the population, $k$. Bühlmann credibility is exactly the linear-Bayes special case of full Bayesian updating: it is the best linear approximation to the true (often intractable) Bayesian posterior mean, and it coincides with it exactly under Normal or certain exponential-family conjugate setups.

Z as an ADE parameter. This is precisely ADE's credibility layer. After forming the Bayesian posterior $p(s\mid e)$ from whatever evidence is available, ADE blends it with a reference/manual distribution $q(s)$ (industry benchmark, class-wide experience, or a regulatory table) using

$p^*(s) = Z\cdot p(s\mid e) + (1-Z)\cdot q(s), \qquad Z\in[0,1].$

$Z=1$ trusts your own posterior fully (appropriate with abundant, directly relevant data); $Z=0$ ignores it entirely and defers to the reference (appropriate for a brand-new risk with no history); intermediate $Z$ is the actuarial workhorse for pricing a class with partial credibility, and $Z$ can itself be set by the Bühlmann formula above, by a Bayesian analysis of the full posterior-vs-reference weighting, or by judgement disclosed as an assumption.

Worked example — reserve credibility. A line of business has 40 claims of own experience suggesting a loss ratio of 68%, while the broader class (2,000 claims) shows 74%. Suppose $k=110$ (calibrated so that roughly 100+ claims are needed for material credibility). Then $Z = 40/(40+110)=0.267$. The credibility-weighted estimate is $0.267(0.68)+0.733(0.74) = 0.182+0.542=0.724$, pulling the raw 68% strongly toward the class mean because the own experience is thin. As claims accumulate to, say, 300, $Z=300/410=0.732$ and the own experience dominates.

Limits. Bühlmann credibility assumes the linear form is a good approximation and that $k$ is estimated reliably — itself a statistical problem (semiparametric or empirical Bayes methods estimate VHM and EVPV from the data). ADE does not resolve that estimation problem; it exposes $Z$ as a transparent, auditable parameter and lets the sensitivity panel show exactly how much the final recommendation depends on the credibility judgement.