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Unformatted text preview: (⇠ x )/⇡ (⇠ ) is easy to compute because Z ( ) cancels out,
as do all the terms in the sum that do not involve x and its neighbors. Since 39 1.7. PROOFS OF THE MAIN THEOREMS*
⇠ x (x) = ⇠ (x).
⇡ (⇠ x )
= exp
⇡ (⇠ ) 2 X ⇠x ⇠y y ⇠x ! If x agrees with k of its four neighbors the ratio is exp( 2(4 2k )). In words
p(x, y ) can be described by saying that we accept the proposed move with
probability 1 if it lowers the energy and with probability ⇡ (y )/⇡ (x) if not.
Example 1.38. Simulated annealing. The MetropolisHastings algorithm
can also be used to minimize complicated functions. Consider for example the
traveling salesman problem, which is to ﬁnd the shortest (or least expensive)
route that allows one to visit all of the cities on a list. In this case the state
space will be lists of cities, x and ⇡ (x) = exp( `(x)) where `(x) is the length
of the tour. The proposal kernel q is chosen to modify the list in some way.
For example, we might move a city to...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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