Lecture 20 Notes

# G represented by factor graphknow only z rx pick n

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: 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 ==&gt; 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...
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