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Discrete Markov random fields form a natural class of models to representimages and spatial data sets. The use of such models is, however, hampered by acomputationally intractable normalising constant. This makes parameterestimation and a fully Bayesian treatment of discrete Markov random fieldsdifficult. We apply approximation theory for pseudo-Boolean functions to binaryMarkov random fields and construct approximations and upper and lower boundsfor the associated computationally intractable normalising constant. As aby-product of this process we also get a partially ordered Markov modelapproximation of the binary Markov random field. We present numerical exampleswith both the pairwise interaction Ising model and with higher-orderinteraction models, showing the quality of our approximations and bounds. Wealso present simulation examples and one real data example demonstrating howthe approximations and bounds can be applied for parameter estimation and tohandle a fully-Bayesian model computationally.

Approximations and bounds for binary Markov random fields