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Markov Random Field - Understanding the Definition

  1. Aug 18, 2007 #1
    First encounter with the term, I'd like some help understanding it. I know there are several approaches, but this one is important to me because I'm trying to understand an article that uses it throughout its text.

    Definition: Let G=(V,E) be a finite (connected) graph and let S be a finite set. A random element X taking values in [tex]S^V[/tex] is said to be a Markov random field if for each [tex]W\subset V[/tex], the conditional distribution of X(W) given X(V\W) depends on X(V\W) only through its values on [tex]\partial W[/tex].

    It goes on to write this mathematically, which I will write down here if you ask me to. My problem is with the phrase "A random element X taking values in...".
    I just want to know from where to where X is. Obviously X takes values in [tex]S^V[/tex], so this is the range of X. What is its domain?
     
  2. jcsd
  3. Aug 18, 2007 #2

    EnumaElish

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    Obviously X's domain is W. E.g. X(w) = 1 if w = Head, X(w) = 0 if w = Tail. Then W = {Head, Tail).
     
  4. Aug 18, 2007 #3
    I don't have W, I have only V and S.

    Did you mean V? So [tex]X:V\rightarrow S^V[/tex]?

    I don't see the sense of that - V doesn't represent states, it is the vertices's set of the graph.

    Could you explain?
     
  5. Aug 18, 2007 #4

    EnumaElish

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    You are right, W is a subset of V; so the domain is V.
     
  6. Aug 19, 2007 #5
    I can't see why. Needless to say I believe you, but I don't understand it.

    You're telling me that [tex]X:V\rightarrow S^V[/tex]? I can't see why, V represents locations, not outcomes in some experiment. I mean, are we assigning to each atom a configuration of the whole system?

    Besides, what is the probability measure on V?
     
    Last edited: Aug 19, 2007
  7. Aug 19, 2007 #6

    EnumaElish

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    I should have differentiated between a vector X of random variables and a set X of random variables. From your definition, X is a set of random variables. Each vertex v of V represents an element of X.

    In the 2-D Isling model, (V,E) looks like:
    ????
    ????
    ????
    ????
    where each ? is a vertex v with two possible values (+,-) and 4 neighbors (North, East, South, West), unless it is sitting on the boundary.

    Had X been defined as a vector, the domain of X would have been the sample space [itex]\Omega[/itex] which underlies the joint probability P(X = x).

    But for a subset U of V, X(U) denotes the image of "those vertices that are included in U." (When X is a set, there is an implicit selection process w/r/t which vertices to include in the set.)

    Although the v's in V have the joint domain [itex]\Omega[/itex], the domain of the set X is defined as V. This emphasizes the selection process. At least that's how I understand it.

    A more complete notation would be to write [itex]\bold X(V(\Omega_V))[/itex] = SV. Although with some misuse of notation one might write [itex]\Omega_V[/itex] for V. This would fold the selection process into the sample space.

    A free e-book can be found at: http://www.ams.org/online_bks/conm1/
     
    Last edited: Aug 20, 2007
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