Statistics and Computing

, Volume 27, Issue 5, pp 1271–1292

Approximate computations for binary Markov random fields and their use in Bayesian models


DOI: 10.1007/s11222-016-9685-7

Cite this article as:
Austad, H.M. & Tjelmeland, H. Stat Comput (2017) 27: 1271. doi:10.1007/s11222-016-9685-7


Discrete Markov random fields form a natural class of models to represent images and spatial datasets. The use of such models is, however, hampered by a computationally intractable normalising constant. This makes parameter estimation and a fully Bayesian treatment of discrete Markov random fields difficult. We apply approximation theory for pseudo-Boolean functions to binary Markov random fields and construct approximations and upper and lower bounds for the associated computationally intractable normalising constant. As a by-product of this process we also get a partially ordered Markov model approximation of the binary Markov random field. We present numerical examples with both the pairwise interaction Ising model and with higher-order interaction models, showing the quality of our approximations and bounds. We also present simulation examples and one real data example demonstrating how the approximations and bounds can be applied for parameter estimation and to handle a fully Bayesian model computationally.


Approximate inference Bayesian analysis Discrete Markov random fields Image analysis Pseudo-Boolean functions Spatial data Variable elimination algorithm 

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.If P& COsloNorway

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