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I have been trying to understand the representation of the joint probability density of Markov random fields in the form of factors of the potential functions. I am finding it difficult to grasp the idea of potential functions and how we are supposed to choose a potential function. Any suggestions on how I should get started. I have gone through the wiki page and some other resources, but still I couldn't grasp the essence of it.

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All potential functions can be written in a log-linear form as described in the wikipedia article. This however may not be that useful, as it requires you to specify a weight for all possible configurations of your clique.

Your choice of potential function depends on the properties of the variables you are modelling. For example, if you are implementing a Kalman filter (which is an autoregressor for continuous variables assuming Gaussian noise), your potential functions are Gaussian. For binary variables (x1, and x2) that should approximate an XOR relationship you could specify the following potential function:

 1/Z * exp(a + b*x1 + b*x2 + c*x1*x2)

where a and b and positive and c is negative. For a very good introduction to probabilistic models I'd recommend Mike Jordan's technical report / book Graphical Models, Exponential Families and Variational Inference or consider taking a look at Chris Bishop's book

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