Lecture 20 Notes

G represented by factor graphknow only z rx pick n

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Unformatted text preview: x)dx f (x)dx Parallel IS Now suppose R(x) is unnormalized (e.g., represented by factor graph)—know only Z R(x) Pick N samples xi from proposal Q(X) If we knew wi = R(xi)/Q(xi), could do IS Instead, set wi = ZR(xi )/Q(xi ) ˆ Parallel IS ˆ E(W ) = = ￿ ￿ ZR(x) Q(x) dx Q(x) ZR(x)dx =Z 1￿ ¯ wi is an unbiased estimate of Z ˆ So, w = Ni Parallel IS ˆ¯ So, wi /w is an estimate of wi, computed without knowing Z Final estimate: ￿ f (x)dx ≈ 1 n ￿ wi ˆ i w g (xi ) ¯ Parallel IS is biased 3 1 / mean(weights) 2.5 E(mean(weights)) 2 1.5 1 0.5 0 0 1 2 mean(weights) 3 ¯ ¯ E (W ) = Z , but E (1/W ) ￿= 1/Z in general 2 1 0 1 2 2 1 0 1 2 Q : (X, Y ) ∼ N (1, 1) θ ∼ U (−π , π ) f (x, y, θ) = Q(x, y, θ)P (o = 0.8 | x, y, θ)/Z 2 1 0 1 2 2 1 0 1 2 Posterior E (X, Y, θ ) = (0.496, 0.350, 0.084) MCMC Integration problem Recall: wanted ￿ f (x)dx = ￿ R(x)g (x)dx And therefore, wanted good importance distribution Q(x) (close to R) Back to high dimensions Picking a good importance distribution is hard in high-D Major contributions to integral can be hidden in small areas ‣ recall, want (R big ==> Q big) Would like to search for areas of high R(x) But searching could bias our estimates Markov-Chain Monte Carlo Design a randomized search procedure M over values of x, which tends to increase R(x) if it is small Run M for a while, take resulting x as a sample Importance distribution Q(x)? Markov-Chain Monte Carlo Design a randomized search procedure M over values of x, which tends to increase R(x) if it is small Run M for a while, take resulting x as a sample Importance distribution Q(x)? ‣ Q = stationary distribution of M… Stationary distribution Run HMM or DBN for a long time; stop at a random point Do this again and again Resulting samples are from stationary distribution Designing a search chain ￿ f (x)dx = ￿ R(x)g (x)dx Would like Q(x) = R(x) ‣ makes importance weight = 1 Turns out we can get this exactly, using Metropolis-Hastings Metropolis-Hastings Way of designin...
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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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