# It is named for two of the authors of the fundamental

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Unformatted text preview: = ⇡ (im+1 )p(im+1 , im ) ⇡ (im ) This shows Ym is a Markov chain with the indicated transition probability. The formula for the transition probability in (1.13), which is called the dual transition probability, may look a little strange, but it is easy to see that it works; i.e., the p(i, j ) 0, and have ˆ X j p(i, j ) = ˆ X ⇡ (j )p(j, i)⇡ (i) = j ⇡ (i) =1 ⇡ (i) since ⇡ p = ⇡ . When ⇡ satisﬁes the detailed balance conditions: ⇡ (i)p(i, j ) = ⇡ (j )p(j, i) the transition probability for the reversed chain, p(i, j ) = ˆ ⇡ (j )p(j, i) = p(i, j ) ⇡ (i) is the same as the original chain. In words, if we make a movie of the Markov chain Xm , 0 m n starting from an initial distribution that satisﬁes the detailed balance condition and watch it backwards (i.e., consider Ym = Xn m for 0 m n), then we see a random process with the same distribution. m. To help explain the concept, 1.6.4 The Metropolis-Hastings algorithm Our next topic is a method for generating samples from a distribution ⇡ (x). It is named for two of the authors of the fundamental papers on the topic. One 37 1.6. SPECIAL EXAMPLES written by Nicholas Metropolis and two married...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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