Bayesian Treatment of Incomplete Discrete Data applied to Mutual Information and Feature Selection

TL;DR

This paper integrates Bayesian methods with EM for incomplete discrete data, deriving efficient approximations for mutual information and applying them to feature selection, demonstrating improved performance over traditional filters.

Contribution

It unifies Bayesian treatment and EM for Dirichlet priors, providing efficient approximations and proving posterior unimodality, with applications to feature selection in incomplete data scenarios.

Findings

Bayesian approach with EM improves feature selection.

Derived efficient approximations for mutual information.

Fast filter outperforms traditional methods on real data.

Abstract

Given the joint chances of a pair of random variables one can compute quantities of interest, like the mutual information. The Bayesian treatment of unknown chances involves computing, from a second order prior distribution and the data likelihood, a posterior distribution of the chances. A common treatment of incomplete data is to assume ignorability and determine the chances by the expectation maximization (EM) algorithm. The two different methods above are well established but typically separated. This paper joins the two approaches in the case of Dirichlet priors, and derives efficient approximations for the mean, mode and the (co)variance of the chances and the mutual information. Furthermore, we prove the unimodality of the posterior distribution, whence the important property of convergence of EM to the global maximum in the chosen framework. These results are applied to the problem of selecting features for incremental learning and naive Bayes classification. A fast filter based on the distribution of mutual information is shown to outperform the traditional filter based on empirical mutual information on a number of incomplete real data sets.