STAT 301 · Year 3 · Semester 1 · 3 credits · Mathematics & Statistics

Bayesian Statistics & Decision Theory

Transform prior belief and empirical loss into optimal actuarial decisions under uncertainty.

Season 1 · 8 episodes

The Uncertainty Ledger

When a £450 million offshore renewable syndicate faces catastrophic early blade failures and sparse claim histories, lead actuary Maya Lin must replace fragile frequentist heuristics with a rigorous, end-to-end Bayesian inference and decision-theoretic architecture before the board shuts down the book.

Protagonist · Maya Lin, ASA, newly appointed Lead Pricing and Reserving Actuary at NorthPoint Specialty Syndicate 4421.

Setting · NorthPoint Specialty Syndicate, Lloyd's of London and Boston; handling high-stakes offshore wind and grid-scale battery storage facilities.

Stakes · A multi-year catastrophic loss corridor threatens syndicate solvency, risking a £60M capital call and the cancellation of NorthPoint's renewable energy book unless Maya can mathematically justify reserves, quantify data acquisition, and prove credibility to regulators.

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Episodes

Season finale: Maya defends a fully unified hierarchical Bayesian decision model before the Lloyd's Franchise Board and PRA regulators, proving how asymmetric loss reserving, optimal drone inspection sampling, and Jewell credibility protect syndicate solvency while preserving the green transition portfolio.

From the lecturer: Welcome to STAT 301. In classical statistics, parameters are fixed constants waiting to be estimated; in Bayesian statistics and decision theory, parameters are random variables capturing your evolving state of knowledge. This paradigm shift provides the mathematical engine behind modern stochastic reserving, dynamic pricing, Bayesian credibility, and enterprise risk management under deep uncertainty. As you watch these films, keep two notebooks open: one for pen-and-paper derivations of conjugate posteriors, predictive integrals, and decision losses, and a digital notebook (Jupyter/Stan/PyMC) for sampling algorithms and diagnostic workflows. Do not merely watch the MCMC traces—re-derive the kernel updates by hand, simulate prior predictive distributions before conditioning on data, and verify that your numerical Bayes decisions match analytical limits under squared-error, absolute, and linex loss functions. While this course directly underpins advanced professional actuarial syllabi (including modern Bayesian credibility in CAS Exam MAS-II and ASTAM reserving frameworks), our primary goal is to forge you into a rigorous probabilistic thinker who quantifies the financial value of imperfect information before committing capital.

Cold open

At 7:14 AM on a wet Monday, Chief Underwriting Officer Marcus Vance drops a confidential dossier on Maya's desk. Three offshore wind turbine gearboxes have suffered catastrophic fatigue fractures off the Dogger Bank within ninety days of commercial operation. With only two years of sparse syndicate operating data, the standard Poisson maximum likelihood frequency estimator yields an absurd premium spike of four hundred percent, threatening to lose the entire North Sea offshore portfolio to competitors before noon.

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Transcript

Formulate Bayes' theorem for continuous parameter spaces and identify the role of the marginal likelihood
Derive posterior distributions for standard conjugate pairs: Beta-Binomial, Gamma-Poisson, and Normal-Normal
Express the posterior mean as a credibility-weighted convex combination of the prior mean and sample mean
Derive and interpret the posterior predictive distribution for future observable claims