Lecture 20 Notes

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

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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’) << 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...
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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.

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