Markov Random Fields
Huizhen Yu, Huizhen Yu · 2010
Markov chains are suitable models for time-series/sequence data. For spatial data, variables can no longer be placed on a line. What would be an analogous model and an analogous Markov property? A natural generalization stems from the following conditional independence property of a Markov chain X1, . . . ,Xn: P(Xj = xj |X−j = x−j) = P(Xj = xj |Xj−1 = xj−1,Xj+1 = xj+1), (1) where X−j = (X1, . . . ,Xj−1,Xj+1, . . . ,Xn). (We proved this in slide 22, Lec. 2.) Consider a collection of random variables {Yv , v ∈ V }, where V consists of “sites” in some space. To generalize property (1) for Y = {Yv} , we introduce the notion of neighbors to site v :