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Unformatted text preview: Problems Primarily of two types: Integration and Optimization Bayesian inference and learning Computing normalization in Bayesian methods p ( y  x ) = p ( y ) p ( x  y ) y p ( y ) p ( x  y ) dy Marginalization: p ( y  x ) = z p ( y , z  x ) dz Expectation: E y  x [ f ( y )] = y f ( y ) p ( y  x ) dy Statistical mechanics: Computing the partition function Z = s exp E ( s ) kT Optimization, Model Selection, etc. Monte Carlo Principle Target density p ( x ) on a highdimensional space Draw i.i.d. samples { x i } n i =1 from p ( x ) Construct empirical point mass function p n ( x ) = 1 n n i =1 x i ( x ) One can approximate integrals/sums by I n ( f ) = 1 n n i =1 f ( x i ) a . s . n I ( f ) = x f ( x ) p ( x ) dx Unbiased estimate I n ( f ) converges by strong law For finite 2 f , central limit theorem implies n ( I n ( f ) I ( f )) = n N (0 , 2 f ) Rejection Sampling Target density p ( x ) is known, but hard to sample Use an easy to sample proposal distribution q ( x ) q ( x ) satisfies p ( x ) Mq ( x ) , M < Algorithm: For i = 1 , , n 1 Sample x i q ( x ) and u U (0 , 1) 2 If u < p ( x i ) Mq ( x i ) , accept x i , else reject Issues: Tricky to bound p ( x ) / q ( x ) with a reasonable constant M If M is too large, acceptance probability is small Rejection Sampling (Contd.) Markov Chains Use a Markov chain to explore the state space Markov chain in a discrete space is a process with p ( x i  x i 1 , . . . , x 1 ) = T ( x i  x i 1 ) A chain is homogenous if T is invariant for all i MC will stabilize into an invariant distribution if 1 Irreducible, transition graph is connected 2 Aperiodic, does not get trapped in cycles Su ffi cient condition to ensure p ( x ) is the invariant distribution...
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 Spring '08
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