Common Voting Rules as Maximum Likelihood Estimators
by Vincent Conitzer and Tuomas Sandholm
Voting is a very general method of preference aggregation. A voting rule takes as input every voter's vote (typically, a ranking of the alternatives), and produces as output either just the winning alternative or an aggregate ranking of the alternatives. One potential view of voting is the following. There exists a ``correct'' outcome (winner/ranking), and each voter's vote corresponds to a noisy perception of this correct outcome. If we are given the noise model, then for any vector of votes, we can compute the maximum likelihood estimate of the correct outcome. This maximum likelihood estimate constitutes a voting rule. In this paper, we ask the following question: For which common voting rules does there exist a noise model such that the rule is the maximum likelihood estimate for that noise model? We require that the votes are drawn independently given the correct outcome (we show that without this restriction, all voting rules have the property). We study the question both for the case where outcomes are winners and for the case where outcomes are rankings. In either case, only some of the common voting rules have the property. Moreover, the sets of rules that satisfy the property are incomparable between the two cases (satisfying the property in the one case does not imply satisfying it in the other case).
A submodular-supermodular procedure with applications to discriminative structure learning
by Mukund Narasimhan and Jeff Bilmes
In their paper, an algorithm for minimizing the difference between two submodular functions is presented, which is based on the concave-convex procedure. Because several commonly used metrics in machine learning (e.g. mutual information) are submodular, the problem of minimizing the difference of two submodular problems arises naturally in many machine learning applications. Two such applications are learning discriminatively structured graphical models and feature selection under computational complexity constraints. The authors also present results on synthetic data indicating that classifiers based on discriminative graphical models learned using this algorithm can significantly outperform classifiers based on generative graphical models.
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Pradeep Ravikumar Last modified: Fri Sep 9 12:02:08 EDT 2005