Bayesian MCMC Methods for Actuarial Applications

MCMC Fundamentals

When posterior distributions lack closed-form solutions, Markov Chain Monte Carlo (MCMC) methods generate samples from the posterior. The Metropolis-Hastings algorithm proposes candidate parameters theta* from a proposal distribution q(theta*|theta_t), then accepts with probability min(1, [pi(theta*|data)*q(theta_t|theta*)] / [pi(theta_t|data)*q(theta*|theta_t)]). The Gibbs sampler is a special case where each parameter is sampled from its full conditional distribution, guaranteeing acceptance. After a burn-in period, the chain's samples approximate the posterior distribution.

Diagnostics and Applications

MCMC diagnostics assess convergence and mixing. Trace plots should show stable, well-mixed chains without trends. The Gelman-Rubin statistic compares within-chain and between-chain variance from multiple chains (R-hat near 1 indicates convergence). Autocorrelation plots guide thinning decisions. Effective sample size accounts for autocorrelation. In actuarial applications, MCMC enables Bayesian fitting of complex loss models, hierarchical credibility models, and stochastic reserving. Posterior predictive checks validate model adequacy. Exam MAS-II tests MCMC algorithm mechanics, convergence diagnostics, and interpretation of posterior summaries for actuarial problems.