Markov Random Field - Understanding the Definition

In summary, the author discusses what a Markov random field is and how it is used in various texts throughout the text. The author asks a question about the domain of a random element and provides a more complete notation.
  • #1
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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?
 
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  • #2
Obviously X's domain is W. E.g. X(w) = 1 if w = Head, X(w) = 0 if w = Tail. Then W = {Head, Tail).
 
  • #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?
 
  • #4
You are right, W is a subset of V; so the domain is V.
 
  • #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?
 
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  • #6
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/
 
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1. What is a Markov Random Field (MRF)?

A Markov Random Field (MRF) is a graphical model used to represent probabilistic relationships between a set of random variables. It is commonly used in fields such as image processing, computer vision, and natural language processing.

2. How is an MRF defined?

An MRF is defined as a set of random variables, each representing a node in a graph. The variables are assumed to follow a Markov property, meaning that the probability of each variable is only dependent on its immediate neighbors in the graph.

3. What is the difference between an MRF and a Markov chain?

An MRF is a generalization of a Markov chain, where the variables can take on more than two values and are not necessarily arranged in a linear sequence. Additionally, the Markov property in an MRF refers to the relationship between variables in a graph, while in a Markov chain it refers to the relationship between variables in a sequence.

4. How are MRFs used in image processing?

In image processing, MRFs are used to model the relationships between pixels in an image. This allows for the incorporation of contextual information, such as neighboring pixels, into the processing and analysis of the image.

5. What are some applications of MRFs?

MRFs have a wide range of applications, including image segmentation, image denoising, object recognition, natural language processing, and speech recognition. They are also commonly used in machine learning algorithms, such as Markov Chain Monte Carlo methods.

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