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531f10MH2

# 531f10MH2 - STAT 531 Metropolis-Hastings(MH Algorithm HM...

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STAT 531: Metropolis-Hastings (MH) Algorithm HM Kim Department of Mathematics and Statistics University of Calgary Fall 2010 1/22

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Choosing a proposal/candidate distribution There are two general approaches: independent chain and random walks chain independent chain the probability of jumping to point y is independent of the current position x of the chain, i.e. q ( y | x ) = g ( y ) the candidate distribution is not symmetric, i.e. g ( y ) 6 = g ( x ) random walks chain the new value y equals to the current value x plus a random variable z q ( y | x ) = g ( z ) Fall 2010 2/22
Recall: Accept-Reject Methods Methods in which we generate a candidate random variable and only accept it subject to passing a test: allow us to simulate from virtually any distribution functional form of the density f interest (=target density) up to a multiplicative constant use a simpler density g (= candidate density) to generate the random variable for which the simulation is actually done f and g are compatible supports (i.e. g ( x ) > 0 when f ( x ) > 0) there is a constant M with f ( x ) g ( x ) M for all x Fall 2010 3/22

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X can be simulated as follows: 1 generate Y g and independently, generate U unif (0 , 1). 2 if U f ( Y ) Mg ( Y ) , then X = Y 3 otherwise Y and U are discarded and start again Fall 2010 4/22
The probability of jumping to point y is independent of the current position x of the chain, i.e. q ( y | x ) = g ( y ). Given x ( t ) , generate Y t g ( y ) take X ( t +1) = ± Y t with probability ρ x ( t ) otherwise . where

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531f10MH2 - STAT 531 Metropolis-Hastings(MH Algorithm HM...

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