Abstract

A random field is the representation of the joint probability distribution for a set of random variables. Markov fields, in particular, have a long standing tradition as the theoretical foundation of many applications in statistical physics and probability. For strictly positive probability densities, a Markov random field is also a Gibbs field, i.e., a random field supplemented with a measure that implies the existence of a regular conditional distribution. Markov random fields have been used in statistical physics, dating back as far as the Ehrenfests. However, their measure theoretical foundations were developed much later by Dobruschin, Lanford and Ruelle, as well as by Hammersley and Clifford. Aside from its enormous theoretical relevance, due to its generality and simplicity, Markov random fields have been used in a broad range of applications in equilibrium and non-equilibrium statistical physics, in non-linear dynamics and ergodic theory. Also in computational molecular biology, ecology, structural biology, computer vision, control theory, complex networks and data science, to name but a few. Often these applications have been inspired by the original statistical physics approaches. Here, we will briefly present a modern introduction to the theory of random fields, later we will explore and discuss some of the recent applications of random fields in physics, biology and data science. Our aim is to highlight the relevance of this powerful theoretical aspect of statistical physics and its relation to the broad success of its many interdisciplinary applications.

Highlights

  • The theory and applications of random fields born out of the fortunate marriage of two simple but deep lines of reasoning

  • That we have presented the fundamentals of MRFs at an introductory level, this may allow to discuss on how these features have impact on their wide range of applications, as the basis for probabilistic graphical models

  • Often MRFs are used in conjunction with inference machines such as Convolutional Neural Networks (CNNs). This is the case of the work by Li and Ping [73] who used a neural conditional random field (NCRF) for metastasis detection from lymph node slide images. Their NCRF approach infers the spatial correlations among neighboring patches via a fully connected conditional MRF incorporated on top of a CNN feature extractor

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Summary

INTRODUCTION

The theory and applications of random fields born out of the fortunate marriage of two simple but deep lines of reasoning. Probabilistic reasoning induced us to think that such multitude of local interactions would be stochastic in nature These two ideas, paramount to statistical mechanics, have been extensively explored and develop into a full theoretical subdiscipline, the theory of random fields. We are aware that by necessity (finiteness), we are leaving out contributions in fields such as sociology (Axelrod models, for instance), finance (volatility maps, Markov switching models, etc.) and others. We believe this panoramic view will make easier for the interested reader to look into these other applications. In Concluding Remarks we will outline some brief concluding remarks

MARKOV RANDOM FIELDS: A THEORETICAL FRAMEWORK
Configuration
Local Characteristics
Cliques
Configuration Potentials
Gibbs Fields
Conditional Independence in Markov Random Fields
Indepence Maps
MARKOV RANDOM FIELDS IN PHYSICS
MRFs in Statistical Mechanics and Mathematical Physics
MRFs in Condensed Matter Physics and Materials Science
Applications of MRFs in Other Areas of Physics
MARKOV RANDOM FIELDS IN BIOLOGY
Applications of MRFs in Biomedical Imaging
Applications of MRFs in Computational Biology and Bioinformatics
Applications of MRFs in Ecology and Other Areas of Biology
MARKOV RANDOM FIELDS IN DATA SCIENCE AND MACHINE LEARNING
Applications of MRFs in Computer Vision and Image Classification
Applications of MRFs in Statistics and Geostatistics
Applications of MRFs in Feature Selection and AI
Applications of MRFs in Computational Linguistics and NLP
Applications of MRFs in the Analysis of Social Networks
Random Fields and Graph Signal Theory
CONCLUDING REMARKS
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