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INTRODUCTION TO STOCHASTIC PROCESSES. MARKOV …
6 February 2017 INTRODUCTION TO STOCHASTIC RANDOM FIELDSD avid A. MeyerDepartment of MathematicsUniversity of California/San DiegoLa Jolla, CA @dajmeyerRecall that in doing Problem 1 on the first midterm we learned that the weather todaydepends upon the weather yesterday and tomorrow, even assuming the MARKOV s make a short excursion to generalize this idea, and in so doing, answer the questionof what we might mean by a MARKOV property for a STOCHASTIC process whose index set issomething other than (a subset of) the AgraphG= (V, E) consists of a setVofverticesand a symmetric relationE V V. Elements ofEare callededges; each is an unordered pair of For anyv V, let theneighborhoodofv,N(v) ={w V|(v, w) E}, ,all vertices inVthat are connected tovby an Given a graphG= (V, E), let{Xv|v V}be a STOCHASTIC process (withindex setV). Suppose that for allv V,Pr(Xv=xv Xw=xw, w V\{v})= Pr(Xv=xv Xw=xw, w N(v)).Then the STOCHASTIC process is aMarkov random field.
Stochastic Processes: Markov random fields David A. Meyer where β > 0 and the partition function, Z(β,n) = X x e−β P i x ix i+1. This is a one-dimensional Ising model with periodic boundary conditions.
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