Bayesian Analysis: Prior, Posterior, and Predictive Distributions
Apply Bayesian statistical methods with prior, posterior, and predictive distributions for Exam MAS-I.
Bayes' Theorem for Distributions
Bayesian analysis combines prior information pi(theta) with observed data through the likelihood L(theta|x) to produce the posterior distribution: pi(theta|x) proportional to L(theta|x) * pi(theta). The posterior summarizes all available information about theta. Conjugate priors simplify computation: when the prior and posterior belong to the same family, updating is straightforward. Key conjugate pairs include Beta-Binomial, Gamma-Poisson, and Normal-Normal. The posterior mean is a credibility-weighted average of the prior mean and the data mean.
Predictive Distributions
The prior predictive distribution integrates over the parameter: f(x) = integral of f(x|theta)*pi(theta)dtheta. The posterior predictive distribution predicts future observations given observed data: f(x_new|x) = integral of f(x_new|theta)*pi(theta|x)dtheta. This incorporates parameter uncertainty into predictions. For a Poisson-Gamma model, the posterior predictive is Negative Binomial. Bayesian credible intervals have a direct probability interpretation: a 95% credible interval contains theta with 95% posterior probability. Exam MAS-I tests conjugate analysis, posterior computation, and predictive distribution derivation.