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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 highD
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 MarkovChain 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)? MarkovChain 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
MetropolisHastings MetropolisHastings
Way of designin...
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This note was uploaded on 01/24/2014 for the course CS 15780 taught by Professor Bryant during the Fall '09 term at Carnegie Mellon.
 Fall '09
 Bryant

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