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Markov Random Field Model in Machine Learning

In the previous article we have learnt about directed graph model called Bayesian graphical Model. Now, in this article we are going to discuss about undirected graph model called Markov's random field model.
Submitted by Bharti Parmar, on March 24, 2019

Markov's Random Fields

Markov random model is a model which use an undirected graph. Undirected graphical models edge represents the potential between two variables, syntactically, Factorization distribution probabilities between variable. In each Individual variable connected with the edge represent a certain clique in the graph; means probability distribution of the variables in the graph can factorize an individual clique potential function.

Just as we had CPDs for Bayesian networks, we have tables to incorporate relations between nodes in Markov networks. However, there are two crucial differences between these tables and CPDs.

Markov Random Field Model diagram 1

Clique in graph theory.it is a subset of vertices of an undirected graph.

P(A, B, C, D, E) α Ф(A,B) Ф (B,C) Ф (B,D) Ф (C,E) Ф (D,E)

Markov Random Field Model diagram 2
Markov Random Field Model diagram 3

Such that: It induces sub graph is complete in every vertices in a clique is adjacent. So, clique in this graph adjust adjacently one by one.


Markov Random Field Model diagram 4

It is some different if we join D,C and B,E clique over here then it is also change its probability.

Markov Random Field Model diagram 5

P(A, B, C, D, E) α Ф(A,B) Ф (B,C,D) Ф (C,D,E)

Some undirected graphic model has Markov Random Field. In MRF certain paths between A and C.

Markov Random Field Model diagram 6
	A -> B -> C
	A -> B -> D -> E -> C

Note that: Markov networks do not need to be acyclic, as was the case with Bayesian networks.

Independence properties such as Markov properties:

  • Any 2 subsets if variables are conditionally independent given a separating subset.
  • If we take 'A' as a subset and 'C' as one subset then there is wore between them. So, there is no way to go between 'A' and 'C' without getting threw the subset. So, we are using (A, B) than B, C, D, E.

Therefore, A and C are separating subsets

  • Any 2 subset of variable are conditionally independent given a separating subset.
  • {B,D}, {B,E} and {B,D,E} are separating subsets.

Conclusion: In this article we will learn about Markov Random Field model, its potential function and its properties.

Reference: Probabilistic Graphical Models Tutorial — Part 1


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