Metropolis-Hastings algorithm

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General Metropolis-Hastings algorithm • Sample values from target distribution generated by taking a random walk through the multidimensional parameter space. – Begins at arbitrary point, specified by user, where P(θ) is non-zero. – At each time step: • Move to new position in parameter space is proposed. • Decide whether or not to accept move to new position. – Proposal distributions can be of various forms. • Goal is to efficiently explore regions of parameter space where P(θ) has greatest mass. • Simplest case: proposal distribution is normal, centered on current position. • Proposed move will typically ybe near present position, with probability of more distant positions decreasing with distance: 0.4 0.35 0.3 025 0.25 P() 0.2 0.15 0.1 0.05 0 -3 -2 -1 0 1 2 3
General Metropolis-Hastings algorithm • Sample values from target distribution generated by taking a random walk through the multidimensional parameter space. – At each time step: • Having generated proposed new position, decision made whether to accept or reject it based on movement rule: p move min P proposed P ,1 current • Random number r generated from uniform interval [0, 1]. • If r is between 0 – p move , move is accepted. – Process repeated. – In the long run: positions visited by the random walk will closely approximate the target distribution. θ 2 θ 1
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 and 10: Metropolis-Hastings algorithm • W
Page 11 and 12: • Algorithm named after: Metropol
Page 13: 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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