### 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) |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*IEEE Transactions on Information Theory*,

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

**On information-theoretic characterizations of markov random fields and subfields.** / Yeung, Raymond W.; Al-Bashabsheh, Ali; Chen, Chao; Chen, Qi; Moulin, Pierre.

Research output: Contribution to journal › Article

*IEEE Transactions on Information Theory*, vol. 65, no. 3, 8444473, pp. 1493-1511. https://doi.org/10.1109/TIT.2018.2866564

}

TY - JOUR

T1 - On information-theoretic characterizations of markov random fields and subfields

AU - Yeung, Raymond W.

AU - Al-Bashabsheh, Ali

AU - Chen, Chao

AU - Chen, Qi

AU - Moulin, Pierre

PY - 2019/3

Y1 - 2019/3

N2 - 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).

AB - 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).

KW - I-Measure

KW - Markov random field

KW - conditional independence

KW - information diagram

KW - subfield

UR - http://www.scopus.com/inward/record.url?scp=85052697986&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052697986&partnerID=8YFLogxK

U2 - 10.1109/TIT.2018.2866564

DO - 10.1109/TIT.2018.2866564

M3 - Article

AN - SCOPUS:85052697986

VL - 65

SP - 1493

EP - 1511

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 3

M1 - 8444473

ER -