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By Edward S. Smith

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X n = si 0 ) . In other words, the chain is equally likely to make a tour through the states si 0 , . . si n in forwards as in backwards order. 7 Markov chain Monte Carlo In this chapter and the next, we consider the following problem: Given a probability distribution π on S = {s1 , . . , sk }, how do we simulate a random object with distribution π? To motivate the problem, we begin with an example. 1: The hard-core model. 1 for the definition of a graph) with vertex set V = {v1 , . . , vk } and edge set E = {e1 , .

J = ρj 1 = τ1,1 τ1,1 = = = = = 1 τ1,1 1 τ1,1 1 τ1,1 1 τ1,1 1 τ1,1 ∞ n=0 ∞ n=1 ∞ P(X n = s j , T1,1 > n) P(X n = s j , T1,1 > n) (27) P(X n = s j , T1,1 > n − 1) (28) n=1 ∞ k P(X n−1 = si , X n = s j , T1,1 > n − 1) n=1 i=1 ∞ k n=1 i=1 ∞ P(X n−1 = si , T1,1 > n − 1)P(X n = s j | X n−1 = si ) (29) k n=1 i=1 Pi, j P(X n−1 = si , T1,1 > n − 1) 32 5 Stationary distributions = = τ1,1 i=1 n=1 ∞ k τ1,1 P(X n−1 = si , T1,1 > n − 1) Pi, j 1 = ∞ k 1 P(X m = si , T1,1 > m) Pi, j i=1 m=0 k i=1 ρi Pi, j τ1,1 k = πi Pi, j (30) i=1 where in lines (27), (28) and (29) we used the assumption that j = 1; note also that (29) uses the fact that the event {T1,1 > n − 1} is determined solely by the variables X 0 , .

K, ρi = ∞ P(X n = si , T1,1 > n) n=0 so that in other words, ρi is the expected number of visits to state i up to time T1,1 − 1. Since the mean return time E[T1,1 ] = τ1,1 is finite, and ρi < τ1,1 , we get that ρi is finite as well. Our candidate for a stationary distribution is π = (π1 , . . , τ1,1 τ1,1 τ1,1 . 1. k πi Pi, j = π j in condition (ii) holds for We first show that the relation i=1 j = 1 (the case j = 1 will be treated separately). ) πj = ρj 1 = τ1,1 τ1,1 = = = = = 1 τ1,1 1 τ1,1 1 τ1,1 1 τ1,1 1 τ1,1 ∞ n=0 ∞ n=1 ∞ P(X n = s j , T1,1 > n) P(X n = s j , T1,1 > n) (27) P(X n = s j , T1,1 > n − 1) (28) n=1 ∞ k P(X n−1 = si , X n = s j , T1,1 > n − 1) n=1 i=1 ∞ k n=1 i=1 ∞ P(X n−1 = si , T1,1 > n − 1)P(X n = s j | X n−1 = si ) (29) k n=1 i=1 Pi, j P(X n−1 = si , T1,1 > n − 1) 32 5 Stationary distributions = = τ1,1 i=1 n=1 ∞ k τ1,1 P(X n−1 = si , T1,1 > n − 1) Pi, j 1 = ∞ k 1 P(X m = si , T1,1 > m) Pi, j i=1 m=0 k i=1 ρi Pi, j τ1,1 k = πi Pi, j (30) i=1 where in lines (27), (28) and (29) we used the assumption that j = 1; note also that (29) uses the fact that the event {T1,1 > n − 1} is determined solely by the variables X 0 , .

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