Lecture 20 Notes

# Rx move to x if rx rx stay at x in intermediate cases

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: g chain w/ Q(x) = R(x) Basic strategy: start from arbitrary x Repeatedly tweak x to get x’ If R(x’) ! R(x), move to x’ If R(x’) &lt;&lt; R(x), stay at x In intermediate cases, randomize Proposal distribution Left open: what does “tweak” mean? Parameter of MH: Q(x’ | x) ‣ one-step proposal distribution Good proposals explore quickly, but remain in regions of high R(x) Optimal proposal? MH algorithm Sample x’ ~ Q(x’ | x) R(x￿ ) Q(x￿ | x) Compute p = R(x) Q(x | x￿ ) With probability min(1, p), set x := x’ Repeat for T steps; sample is x1, …, xT (will usually contain duplicates) MH algorithm Sample x’ ~ Q(x’ | x) note: we don’t need to know Z R(x￿ ) Q(x￿ | x) Compute p = R(x) Q(x | x￿ ) With probability min(1, p), set x := x’ Repeat for T steps; sample is x1, …, xT (will usually contain duplicates) MH example 1 0.5 0 0.5 1 1 0.5 0 0.5 1 Acceptance rate Moving to new x’ is accepting Want acceptance rate (avg p) to be large, so we don’t get big runs of the same x Want Q(x’ | x) to move long distances (to explore quickly) Tension between Q and P(accept): R(x￿ ) Q(x￿ | x) p= R(x) Q(x | x￿ ) Mixing rate, mixing time If we pick a good proposal, we will move rapidly around domain of R(x) After a short time, won’t be able to tell where we started—we have reached stationary dist’n This is short mixing time = # steps until we can’t tell which starting point we used Mixing rate = 1 / (mixing time) MH estimate Once we have our samples x1, x2, … Optional: discard initial “burn-in” range ‣ allows time to reach stationary dist’n Estimated integral: N ￿ 1 g (xi ) N i=1 In example g(x) = x2 True E(g(X)) = 0.28… Proposal: Q(x￿ | x) = N (x￿ | x, 0.252 I ) Acceptance rate 55–60% After 1000 samples, minus burn-in of 100: final final final final final estimate estimate estimate estimate estimate 0.282361 0.271167 0.322270 0.306541 0.308716 Gibbs sampler Special case of MH Divide X into blocks of r.v.s B(1), B(2...
View Full Document

## This note was uploaded on 01/24/2014 for the course CS 15-780 taught by Professor Bryant during the Fall '09 term at Carnegie Mellon.

Ask a homework question - tutors are online