Reciprocal properties of random fields on undirected graphs

Abstract

AbstractWe clarify and refine the definition of a reciprocal random field on an undirected graph, with the reciprocal chain as a special case, by introducing four new properties: the factorizing, global, local, and pairwise reciprocal properties, in decreasing order of strength, with respect to a set of nodes $\delta$ . They reduce to the better-known Markov properties if $\delta$ is the empty set, or, with the exception of the local property, if $\delta$ is a complete set. Conditions for each reciprocal property to imply the next stronger property are derived, and it is shown that, conditionally on the values at a set of nodes $\delta_0$ , all four properties are preserved for the subgraph induced by the remaining nodes, with respect to the node set $\delta\setminus\delta_0$ . We note that many of the above results are new even for reciprocal chains.