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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)
‣ onestep 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 “burnin” 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 burnin 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 15780 taught by Professor Bryant during the Fall '09 term at Carnegie Mellon.
 Fall '09
 Bryant

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