Bayesian Statistics

Prior and posterior distributions, conjugate families, MCMC, and Bayesian estimation.

Key Concepts

• •Bayes theorem: posterior proportional to likelihood times prior
• •Prior distributions: informative, non-informative, and conjugate priors
• •Conjugate families: beta-binomial, gamma-Poisson, normal-normal
• •Posterior distributions: analytical solutions for conjugate families
• •Predictive distributions: averaging over parameter uncertainty
• •Loss functions: squared error, absolute error, and zero-one loss
• •Bayes estimators: posterior mean, median, and mode under different losses
• •Credibility as Bayesian estimation: the connection between credibility and posterior means
• •MCMC methods: Gibbs sampling and Metropolis-Hastings for complex posteriors
• •Bayesian model comparison: Bayes factors and posterior model probabilities

Study Tips

• 1.Master conjugate prior calculations, as these appear frequently on exams.
• 2.Practice deriving posterior distributions step by step.
• 3.Understand the connection between Bayesian estimation and credibility theory.
• 4.Work through predictive distribution problems for Poisson-gamma and other conjugate pairs.
• 5.Focus on analytical solutions for exam preparation rather than MCMC implementation.

Related Exam Resources