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This question is about graphical modeling of joint probability functions, Markovian property and Markov random fields.

Suppose we have an undirected graph G where each node represents a random variable and an edge between two nodes says that there is a probabilistic relation in between them. I want to model the joint probability of these variables and to simply simplify things I assume that the graph/nodes have Markovian property. This should let me write the joint as a factor factorization over local" clique potentials.

In C. Bishop's Pattern Recog. and Machine Learning book, Chapter 8, pp. 386 (pdf of the chapter), it is said that the joint distribution is written as a product of potential functions over the maximal cliques of the graph:

$p(x) = \frac{1}{Z} \prod_C \psi_C(x_C)$ (eq. 8.39)

However, in Stan Li's book on MRFs, he says this factorization is done over all possible cliques of the graph: see equation (1.26) and (1.27) in http://www.nlpr.ia.ac.cn/users/szli/MRF_Book/Chapter_1/node12.html#SECTION00323000000000000000.

Stan Li's explanation makes more sense to me. Which one do you think is correct? Or, might they be just different wording of the same fact? Any help would be appreciated.

This question is about graphical modeling of joint probability functions, Markovian property and Markov random fields.

Suppose we have an undirected graph G where each node represents a random variable and an edge between two nodes says that there is a probabilistic relation in between them. I want to model the joint probability of these variables and to simply things I assume that the graph/nodes have Markovian property. This should let me write the joint as a factor over local" clique potentials.

In C. Bishop's Pattern Recog. and Machine Learning book, Chapter 8, pp. 386 (pdf of the chapter), it is said that the joint distribution is written as a product of potential functions over the maximal cliques of the graph:

$p(x) = \frac{1}{Z} \prod prod_C \psi_C(x_C)$ (eq. 8.39)

However, in Stan Li's book on MRFs, he says this factorization is done over all possible cliques of the graph: see equation (1.26) and (1.27) in http://www.nlpr.ia.ac.cn/users/szli/MRF_Book/Chapter_1/node12.html#SECTION00323000000000000000.

Stan Li's explanation makes more sense to me. Which one do you think is correct? Or, might they be just different wording of the same fact? Any help would be appreciated.

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How to factorize the joint probability of an arbitrary graph whose nodes are random variables?

This question is about graphical modeling of joint probability functions, Markovian property and Markov random fields.

Suppose we have an undirected graph G where each node represents a random variable and an edge between two nodes says that there is a probabilistic relation in between them. I want to model the joint probability of these variables and to simply things I assume that the graph/nodes have Markovian property. This should let me write the joint as a factor over local" clique potentials.

In C. Bishop's Pattern Recog. and Machine Learning book, Chapter 8, pp. 386 (pdf of the chapter), it is said that the joint distribution is written as a product of potential functions over the maximal cliques of the graph:

$p(x) = \frac{1}{Z} \prod \psi_C(x_C)$