Markov Property in Generative Classifiers

Generative classifiers are a wide class of machine learning models that consist in estimating the joint probability distributions over the predictor and class variables. From the estimated distribution a decision can be made over the class variable given the values of the predictors. Algebraic and geometric methods can be valuable tools in dealing with discrete probabilities as graphical models(Garcia et al., 2005; Settimi and Smith, 1998), contingency tables and exponential families (Diaconis and Sturmfels, 1995; Fienberg and Gilbert, 1970). Varando et al. (2015) have studied the decision functions induced by a large class of generative classifiers based on Bayesian networks, extending the results of Minsky (1961); Peot (1996) and Jaeger (2003). Ling and Zhang (2002) have described the complexity of Bayesian network classifiers linking the graph structure with the maximum order of the XORs that are representable by the corresponding classifier. In this article we develop a framework to study generative binary classifiers, over categorical predictors, under conditional independences.