Conditional information and definition of neighbor in categorical random fields

TL;DR

This paper investigates the limitations of the traditional neighbor definition in Markov random fields when the joint distribution isn't positive, proposing alternative concepts for such cases.

Contribution

It introduces new conditions under which additional site information affects beliefs and when some sites' information is equivalent to all others, extending neighbor concepts.

Findings

Traditional neighbor definition is not well-defined without positivity.

Conditions identified for when extra site information influences beliefs.

Alternative neighbor concepts proposed for non-positive joint distributions.

Abstract

We show that the definition of neighbor in Markov random fields as defined by Besag (1974) when the joint distribution of the sites is not positive is not well-defined. In a random field with finite number of sites we study the conditions under which giving the value at extra sites will change the belief of an agent about one site. Also the conditions under which the information from some sites is equivalent to giving the value at all other sites is studied. These concepts provide an alternative to the concept of neighbor for general case where the positivity condition of the joint does not hold.