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I am learning about some of the common applications of Markov random fields (a.k.a. undirected graphical models) to data science. A common feature of many applications I have read about is that the number of variables in the model is relatively large (e.g. in applications to computer vision or NLP). For educational purposes, I am interested in the following question:

Are there nice examples of MRF models where the number of variables in the model is very small (i.e. $<10$)?

Ideally, such examples would have some practical use. I would like more than just a toy model graph labelled with some variables (a real model based on a real data set is what I'm looking for).

As a follow up:

If no, why not? Is there something about MRF that makes modelling small numbers of variables uninteresting? Are other types of models preferred in that case for some reason? If yes, please elaborate on the example.

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I cannot think about a small MRF with a practical use. Typically they are just toy examples to help understand the definitions such as in this document (Section 2).

Two points to answer your follow up questions:

-If you have a very small number of variables, you are likely to have a much better understanding of the relations between them. Then you might prefer a directed graphical model, in which it is easier to insert your knowledge (in the form of conditional probabilities) than it is for an undirected graphical model where you need to write potentials for the energy function.

-The partition function becomes rapidly intractable in a MRF, even for relatively small ones. This forces to perform simulations and inference through stochastic simulations such as Gibbs sampling. These methods are not reliable for such small-sized MRF.

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