Mutual information has been successfully adopted in filter feature-selection methods to assess both the relevancy of a subset of features in predicting the target variable and the redundancy with respect to other variables. However, existing algorithms are mostly heuristic and do not offer any guarantee on the proposed solution. In this paper, we provide novel theoretical results showing that conditional mutual information naturally arises when bounding the ideal regression classification errors achieved by different subsets of features. Leveraging on these insights, we propose a novel stopping condition for backward and forward greedy methods which ensures that the ideal prediction error using the selected feature subset remains bounded by a user-specified threshold. We provide numerical simulations to support our theoretical claims and compare to common heuristic methods.
A related theoretical study of feature selection via MI has been recently proposed by @cite_25 . The authors show that the problem of finding the minimal feature subset such that the conditional likelihood of the targets is maximized is equivalent to minimizing the CMI. Based on this result, common heuristics for information-theoretic feature selection can be seen as iteratively maximizing the conditional likelihood. Similarly, we show a connection between the CMI and the optimal prediction error. Differently from @cite_25 , we additionally propose a novel stopping condition that is well motivated by our theoretical findings.
@cite_25
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We present a unifying framework for information theoretic feature selection, bringing almost two decades of research on heuristic filter criteria under a single theoretical interpretation. This is in response to the question: "what are the implicit statistical assumptions of feature selection criteria based on mutual information?". To answer this, we adopt a different strategy than is usual in the feature selection literature--instead of trying to define a criterion, we derive one, directly from a clearly specified objective function: the conditional likelihood of the training labels. While many hand-designed heuristic criteria try to optimize a definition of feature 'relevancy' and 'redundancy', our approach leads to a probabilistic framework which naturally incorporates these concepts. As a result we can unify the numerous criteria published over the last two decades, and show them to be low-order approximations to the exact (but intractable) optimisation problem. The primary contribution is to show that common heuristics for information based feature selection (including Markov Blanket algorithms as a special case) are approximate iterative maximisers of the conditional likelihood. A large empirical study provides strong evidence to favour certain classes of criteria, in particular those that balance the relative size of the relevancy redundancy terms. Overall we conclude that the JMI criterion (Yang and Moody, 1999; , 2008) provides the best tradeoff in terms of accuracy, stability, and flexibility with small data samples.