Learning Structure of Partial Markov Random Field via Partitioned Ratio

A new concept, \emph{partitioned ratio} is proposed to find the partial connectivity of the Markov random field. First we argue this partitioned ratio has a profound link with the Markov properties of random variables via its factorization. Specifically, partitioned ratio may be further decomposed into \emph{Bridges}, a novel subgraph structure, capturing the partial connectivity of the Markov random field, which can be roughly considered as the "link" structure between two partitions. Second, a simple one-shot optimization is illustrated to learn the sparse factorizations of partitioned ratio efficiently, regardless the Gaussianity of the joint distribution or the marginal distributions. Third, we show the sufficient conditions for the proposed algorithm recovering the correct \emph{pairwise} bridge factorizations.