  Let $X_i, i \in V$ 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 $X_i,
i \in V'$ 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 $X_i, i \in V$ form a Markov chain, it is known that the
$I$-Measure is always nonnegative and the information diagram assumes a very
special structure Kawabata and Yeung (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 in Yeung, Lee, and Ye (2002).
