030507 - Lecture 12 Recall Metropolis Algorithm S big...

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Lecture 12: 03/05/2007 Recall: Metropolis Algorithm S: big discrete set of states A probability distribution on S: ( )∝ ∈( ) p x e x kT Algorithm constructs a Markov Chain using “proposal followed by accept/reject.” Start with symmetric Q ij ; Q ij =prob{given I, propose j} Accept j as next state with probability ( , )= < ( ) -( - ) > ( ) α i j 1 if ϵj ϵ i e ϵj ϵi kT if ϵj ϵ i Overall transition probabilities for Markov chain are: P(I,j)=Q(I,j)α(I,j) Recall detailed balance ideas given a probabilitiy distribution π on S, their πis stationary distribution for MC with transition probs P(I,j) if , = , , πiPi j Pj iπjfor all i j (Detailed balance conditions) Claim : with P(I,j)= Q(I,j)α(I,j), defined as above, stationary distribution is precisely π * , where *= - - πi e ϵikTj Se ϵjkT Ie, “π * =P” Proof : Check detailed balance conditions: Case 1: > ( ) ϵi ϵ j . Then P(I,j)=Q(I,j) and P(j,i)=Q(j,i) -( - ) e ϵi ϵj kT Since Q(j,i)=Q(I,j), this says P(I,j)=Q(I,j)=Q(j,i)=P(j,i) ( - ) e ϵi ϵj kT Note: ( - ) = ( ) ( )= * * e ϵi ϵj kT p j p i πj πi Hence: * , = , * πi Pi j Pj iπj Bottom Line : π * , the Boltzmann distribution, is the (unique*) stationary distribution for Markov chain (easy to argue similarly when > . ϵj ϵi *unique because, by construction, the Markov chain is generic -every state positively recurrent, only one recurrence class, hence only one stationary distribution.
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