# To obtain many samples we output the state at widely

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Unformatted text preview: couples with last names Rosenbluth and Teller (1953) and the other by W.K. Hastings (1970). This a very useful tool for computing posterior distributons in Bayesian statistics (Tierney (1994)), reconstructing images (Geman and Geman (1984)), and investigating complicated models in statistical physics (Hammersley and Handscomb (1964)). It would take us too far aﬁeld to describe these applications, so we will content ourselves to describe the simple idea that is the key to the method. We begin with a Markov chain q (x, y ) that is the proposed jump distribution. A move is accepted with probability ⇢ ⇡ (y )q (y, x) r(x, y ) = min ,1 ⇡ (x)q (x, y ) so the transition probability p(x, y ) = q (x, y )r(x, y ) To check that ⇡ satisﬁes the detailed balance condition we can suppose that ⇡ (y )q (y, x) &gt; ⇡ (x)q (x, y ). In this case ⇡ (x)p(x, y ) = ⇡ (x)q (x, y ) · 1 ⇡ (y )p(y, x) = ⇡ (y )q (y, x) ⇡ (x)q (x, y ) = ⇡ (x)q (x, y ) ⇡ (y )q (y, x) To generate one sample from ⇡ (x) we run the chain for a long tim...
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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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