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Exploiting Parameter Related Domain Knowledge for Learning in Graphical Models

Building accurate models from a small amount of available training data can sometimes prove to be a great challenge. Expert domain knowledge can often be used to alleviate this burden. Parameter Sharing is one such important form of domain knowledge. Graphical models like HMMs, DBNs and Module Networks use different forms of Parameter Sharing to reduce the variance in the parameter estimates. The goal of this paper is to present a theoretical approach for learning in presence of several other types of Parameter Related Domain Knowledge that go beyond the ones in the above models. First, we introduce a General Parameter Sharing Framework that describes the models just mentioned, but allows for much finer grained parameter sharing assumptions. In this framework, we present sound procedures for parameter learning from both a Frequentist and a Bayesian point of view, from both complete and incomplete data, in the case where a domain expert specifies in advance the structure of the graphical model, and the subsets of parameters to be shared. Second, we describe a hierarchical extension of this framework based on Parameter Sharing Trees. Finally we present algorithms for using domain knowledge that specifies that certain groups of parameters share certain properties. In particular, we consider two kinds of constraints: first kind states certain groups of parameters share the same aggregate probability mass and second kind states the ratio of the parameters is preserved (shared) in several groups. As an example, we derive a novel form of parameter sharing for Bayesian Multinetworks.