metropolis

metropolis - Winter 2010 Pstat160A Metropolis Algorithm...

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Winter 2010 Pstat160A Hand-out MC #8 Metropolis Algorithm 1. Setting Suppose that X is a random variable (typically a vector), with values in a countable set S . P ( X = i ) = p ( i ) , i S We are interested in the expected value for a function h , E ± h ( X ) ² or in percentiles, e.g., find a such that P ( X a ) = . 9, for instance to construct a confidence interval. In some circumstances, an analytical or numerical solution is not available, and even a direct simulation is not feasible. Typically, the reason is that the distribution is known only up to a constant, that is p ( i ) = Kf ( i ) K > 0 is unknown. If the state space is very large, it is impossible to evaluate this constant by direct summation. That is it is not practically feasible to find K by i S f ( i ) = 1 /K . The Metropolis procedure constructs a Markov chain whose stationary distribution π is the desired p ( i ), using only the known function f ( i ). 2. Algorithm The basic ingredient is the function f ( i ) above and a transition matrix Q = ( Q ij ) from which we can easily simulate, for example a random walk on a graph with vertices S . Assume Q ii = 0 and define α ( i,j ) = min ³ 1 , f ( j ) Q ji f ( i ) Q ij ´ (1) P ij = Q ij α ( i,j ) if i 6 = j (2) P ii = 1 - X j 6 = i P ij (3) Let X n be a Markov chain with transition matrix
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metropolis - Winter 2010 Pstat160A Metropolis Algorithm...

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