### Abstract

Let X _{i} , i ϵ V form a Markov random field (MRF) represented by an undirected graph G = (V,E) , and V′ be a subset of V. We determine the smallest graph that can always represent the subfield X _{i} , i ϵ V′ as an MRF. Based on this result, we obtain a necessary and sufficient condition for a subfield of a Markov tree to be also a Markov tree. When G is a path so that X _{i} , i ϵ V form a Markov chain, it is known that the I -Measure is always nonnegative (Kawabata and Yeung in 1992). We prove that Markov chain is essentially the only MRF such that the I -Measure is always nonnegative. By applying our characterization of the smallest graph representation of a subfield of an MRF, we develop a recursive approach for constructing information diagrams for MRFs. Our work is built on the set-theoretic characterization of an MRF (Yeung et al. in 2002).

Original language | English (US) |
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Article number | 8444473 |

Pages (from-to) | 1493-1511 |

Number of pages | 19 |

Journal | IEEE Transactions on Information Theory |

Volume | 65 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2019 |

### Keywords

- I-Measure
- Markov random field
- conditional independence
- information diagram
- subfield

### ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences

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## Cite this

*IEEE Transactions on Information Theory*,

*65*(3), 1493-1511. [8444473]. https://doi.org/10.1109/TIT.2018.2866564