Mixed markov fields
David B. Mumford, Artur Yakolevich Fridman · 2000
Markov random fields (MRF's) have proven to be a powerful tool for organizing and structuring the acquired knowledge. In a MRF, two nodes are connected if the corresponding variables are directly dependent. In particular, one must identify every pair of variables that could ever be directly dependent on each other and connect the corresponding nodes. This rule may lead to excessively dense networks, in which it is difficult to perform inference. In this thesis, we introduce a class of probability distributions on graphs, called mixed Markov fields, which extend MRF's by allowing a variable to be statically linked to another variable or temporarily dynamically bound to it. The network of a mixed Markov field may undergo local changes during the inference process. While the concept of retractable dynamic links between variables is undeniably appealing, there arises the question of consistency between the joint distribution of the model, on one hand, and its local characteristics, on the other hand. We settle this issue by proving an analog of the Hammersley Clifford equivalence theorem for mixed Markov fields. The new model comes equipped with an MCMC-type algorithm for approximate inference. In this thesis, we establish the algorithm's theoretical and computational feasibility and illustrate the inference process with a number of numerical experiments.