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Lecture 20 Notes

# Lecture 20 Notes - 15-780 Grad AI Lecture 20 Monte Carlo...

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15-780: Grad AI Lecture 20: Monte Carlo methods, Bayesian learning Geoff Gordon (this lecture) Tuomas Sandholm TAs Erik Zawadzki, Abe Othman

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Admin Reminder: midterm March 29 Tuomas’s review session tomorrow, mine yesterday Reminder: project milestone reports due March 31
Review: factor graphs Undirected, bipartite graph one set of nodes represents variables other set represents factors in probability distribution—tables of nonnegative numbers need to compute normalizer in order to do anything useful Can convert back and forth to Bayes nets Hard v. soft constraints

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Review: factor graphs Graphical test for independence different results from Bayes net, even if we are representing the same distribution Inference by dynamic programming instantiate evidence, eliminate nuisance nodes, normalize, answer query elimination order matters treewidth Relation to logic
Review: HMMs, DBNs Inference over time same graphical template repeated once for each time step—conceptually inFnite Inference: forward- backward algorithm (special case of belief propagation)

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Review: numerical integration Integrate a difFcult function over a high- dimensional volume narrow, tall peaks contribute most of the integral—difFcult search problem Central problem for approximate inference e.g., computing normalizing constant in a factor graph
Uniform sampling ï 1 ï 0.5 0 0.5 1 0 10 20 30 40 50 60 70 2 N ° i f ( x i )

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Importance sampling ï 1 ï 0.5 0 0.5 1 0 10 20 30 40 50 60 70
Variance How does this help us control variance? Suppose f big ==> Q big And Q small ==> f small Then h = f/Q never gets too big Variance of each sample is lower ==> need fewer samples A good Q makes a good IS

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Importance sampling, part II Suppose f ( x )= R ( x ) g ( x ) ° f ( x ) dx = ° R ( x ) g ( x ) dx = E R [ g ( x )]
Importance sampling, part II Use importance sampling w/ proposal Q(X): Pick N samples x i from Q(X) Average w i g(x i ), where w i = R(x i )/Q(x i ) is importance weight E Q ( Wg ( X )) = ° Q ( x ) R ( x ) Q ( x ) g ( x ) = ° R ( x ) g ( x ) dx = ° f ( x ) dx

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Parallel IS Now suppose R(x) is unnormalized (e.g., represented by factor graph)—know only Z R(x) Pick N samples x i from proposal Q(X) If we knew w i = R(x i )/Q(x i ), could do IS Instead, set ˆ w i = ZR ( x i ) /Q ( x i )
Parallel IS So, is an unbiased estimate of Z ¯ w = 1 N ° i ˆ w i E ( ˆ W )= ° Q ( x ) ZR ( x ) Q ( x ) dx = ° ( x ) dx = Z

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Parallel IS So, is an estimate of w i , computed without knowing Z Final estimate: ˆ w i / ¯ w f ( x ) dx 1 n ± i ˆ w i ¯ w g ( x i )
Parallel IS is biased 0 1 2 3 0 0.5 1 1.5 2 2.5 3 mean(weights) 1 / mean(weights) E(mean(weights)) E ( ¯ W )= Z , but E (1 / ¯ W ) ° =1 /Z in general

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ï 2 ï 1 0 1 2 ï 2 ï 1 0 1 2 Q :( X,Y ) N (1 , 1) θ U ( π , π ) f ( x,y, θ )= Q ( θ ) P ( o =0 . 8 | θ ) /Z
ï 2 ï 1 0 1 2 ï 2 ï 1 0 1 2 Posterior E ( X,Y, θ ) = (0 . 496 , 0 . 350 , 0 . 084)

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MCMC
Integration problem Recall: wanted And therefore, wanted good importance distribution Q(x) (close to R) f ( x ) dx = R ( x ) g ( x ) dx

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Lecture 20 Notes - 15-780 Grad AI Lecture 20 Monte Carlo...

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