Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 26J / a / Solution

In Bayesian inference, a finite parameter space has posterior probabilities $\pi (\theta )=w(\theta )/Z$, with $w(\theta )$ the product of the prior probability and likelihood and $Z={\sum }_{\theta }w(\theta )$ the evidence. Although each weight may be easy to evaluate, summing a large parameter space can be impractical. The Metropolis–Hastings algorithm constructs a Markov chain with posterior stationary distribution, allowing posterior expectations, probabilities and credible regions to be estimated from its trajectory. Its acceptance ratios cancel $Z$, so direct enumeration of the normalizing constant is unnecessary. Posterior sampling alone does not automatically provide the evidence for comparing models.