Metropolis-Hastings algorithm

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Metropolis-Hastings algorithm • Note: target distribution ib i does not need to be normalized. – Based on ratio: P(θ proposed ) / P(θ current ) . – Useful when target distribution P(θ) is a posterior distribution p D| p . proportional to pD| p Bayes’ rule: p | D p D p | D pD| p – Need only evaluate the product p D| p , not the separate likelihoods and priors. – By evaluating p D | p , can generate random representative values from the posterior distribution. – Don’t need to evaluate evidence, pD. • Can do Baysian inference with rich and complex models.
• Algorithm named after: Metropolis-Hastings algorithm – Nicholas Metropolis (et al., 1953), physicist who was first author (of 5) on paper that proposed it. (Other coauthors actually did more work.) – W. Keith Hastings (1970), mathematician who extended to the general case. • Commonly used Markov-chain Monte Carlo (MCMC) method. • Why Markov • Markov process: process in which the probability of passing from a current state to another particular state is constant, and depends d only on the current state. • Markov chain: probabilistic instance of a Markov process based on a random walk through the probabilistic decisions.
Page 1 and 2: Multiple parameters • Problem: Pi
Page 3 and 4: Monte Carlo approach • Basic goal
Page 5 and 6: Metropolis-Hastings algorithm • I
Page 7 and 8: Metropolis-Hastings algorithm • P
Page 9: Metropolis-Hastings algorithm • W
Page 13 and 14: General Metropolis-Hastings algorit
Page 15 and 16: General Metropolis-Hastings algorit
Page 17 and 18: Efficiency, “burn-in”, and conv
Page 19: Efficiency, “burn-in”, and conv

Metropolis-Hastings algorithm

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