Insurance pricing and reserving are fundamentally prediction problems: estimating the expected cost of future claims from historical data, policyholder characteristics, and expert judgment about emerging risks. Bayesian methods have been embedded in actuarial practice since long before they became fashionable in other fields — credibility theory, the actuary's tool for blending individual and group experience, is one of the oldest applications of Bayesian reasoning in any industry.
Credibility Theory
Credibility theory addresses the central actuarial question: how much weight should an individual policyholder's (or group's) experience receive relative to the overall population? A new driver with no claims history should be priced at the population average; a driver with 20 years of claims data should be priced primarily on their individual experience. The credibility formula provides the optimal blend.
Z = n / (n + k), where n = exposure and k = variance ratio
This is the posterior mean under a Normal-Normal Bayesian model:
θᵢ ~ N(μ, τ²), X̄ᵢ | θᵢ ~ N(θᵢ, σ²/n)
The credibility factor Z ranges from 0 (no individual data, rely entirely on the population) to 1 (abundant individual data, ignore the population). This is precisely the partial pooling that Bayesian hierarchical models provide: individual risk parameters are drawn from a population distribution, and the posterior for each individual is a weighted average of their own data and the population mean, with weights determined by the relative precision of each source.
Claims Modeling
Insurance claims have two components: frequency (how many claims?) and severity (how large is each claim?). Bayesian methods model both, typically using Poisson or negative binomial distributions for frequency and gamma, lognormal, or Pareto distributions for severity, with hierarchical priors that allow parameters to vary across rating factors (age, geography, vehicle type) while sharing information across groups. The compound distribution of total claims — obtained by combining frequency and severity — provides the posterior predictive distribution of total losses.
Insurers must reserve capital for claims that have occurred but not yet been fully paid (IBNR — Incurred But Not Reported). The chain-ladder method, the standard actuarial technique, has a Bayesian interpretation as inference in a multiplicative model with improper priors. Bayesian loss reserving replaces the chain-ladder's point estimates with full predictive distributions of reserves, quantifying the uncertainty that regulators and auditors increasingly demand. Models like the Bayesian over-dispersed Poisson and correlated chain-ladder enable correlations across accident years and development periods.
Catastrophe Risk and Reinsurance
Catastrophe risk — from hurricanes, earthquakes, floods, and pandemics — involves rare events with limited historical data and potentially enormous losses. Bayesian methods combine physical models (hurricane track simulations, seismic hazard models) with historical loss data and expert judgment to estimate exceedance probability curves. Reinsurance pricing, which depends on tail risk estimates, benefits particularly from the Bayesian propagation of uncertainty: the price of a reinsurance contract should reflect not just the expected loss but the uncertainty in the expected loss.
Mortality Modeling and Life Insurance
Bayesian mortality models estimate death rates by age, sex, and cohort, with priors that enforce smoothness across ages and coherent trends over time. The Lee-Carter model and its extensions, fitted in a Bayesian framework, provide posterior predictive distributions for future mortality rates and life expectancies. These drive the pricing of annuities, life insurance, and pension liabilities, where the financial consequences of misestimating future mortality improvements can be enormous.
"Credibility theory is the actuary's Bayesian theorem, predating the modern Bayesian revival by decades. Actuaries were Bayesians before it was fashionable — because the data demanded it." — Stuart Klugman, on the Bayesian foundations of actuarial science
Current Frontiers
Telematics data from connected vehicles enables real-time Bayesian updating of driver risk profiles. Bayesian methods for cyber risk — an emerging peril with virtually no historical loss data — rely heavily on priors from expert elicitation and scenario analysis. Climate change adaptation in insurance pricing uses Bayesian integration of climate projections with loss models. And the application of Bayesian causal inference to insurance — distinguishing selection effects from true risk differences — informs fairer pricing and regulatory compliance.