1 / 10

Metropolis- Hastings Algorithm

Inference. Probabilistic Graphical Models. Sampling Methods. Metropolis- Hastings Algorithm. Reversible Chains. Theorem: If detailed balance holds, and T is regular, then T has a unique stationary distribution  Proof:. Metropolis Hastings Chain. Proposal distribution Q( x  x ’).

sybill-stout
Download Presentation

Metropolis- Hastings Algorithm

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Inference Probabilistic Graphical Models Sampling Methods Metropolis-HastingsAlgorithm

  2. Reversible Chains Theorem: If detailed balance holds, and T is regular, then T has a unique stationary distribution  Proof:

  3. Metropolis Hastings Chain Proposal distribution Q(x x’) • At each state x, sample x’ from Q(x x’) • Accept proposal with probability A(x x’) • If proposal accepted, move to x’ • Otherwise stay at x Acceptance probability: A(x x’) T(x x’) = if x’x Q(x x’) A(x x’) T(x x) = Q(x x) + x’xQ(x x’) (1-A(x x’))

  4. Acceptance Probability

  5. Choice of Q • Q must be reversible: • Q(x x’) > 0  Q(x’  x) > 0 • Opposing forces • Q should try to spread out, to improve mixing • But then acceptance probability often low

  6. MCMC for Matching Xi = j if i matched to j if every Xi has different value otherwise

  7. MH for Matching:Augmenting Path 1) randomly pick one variable Xi 2) sample Xi, pretending that all values are available 3) pick the variable whose assignment was taken (conflict), and return to step 2 • When step 2 creates no conflict, modify assignment to flip augmenting path

  8. Example Results Gibbs MH proposal 1 MH proposal 2

  9. Summary • MH is a general framework for building Markov chains with a particular stationary distribution • Requires a proposal distribution • Acceptance computed via detailed balance • Tremendous flexibility in designing proposal distributions that explore the space quickly • But proposal distribution makes a big difference • and finding a good one is not always easy

  10. END END END

More Related