On the Relationship between the Posterior and Optimal Similarity

TL;DR

This paper explores the properties of Bayesian similarity measures in classification, demonstrating how they relate to class posterior reconstruction, optimal classification, and their limitations in multitask learning scenarios.

Contribution

It provides methods to reconstruct class posteriors from Bayesian similarity and constructs classifiers surpassing nearest neighbor performance, also analyzing limitations in multitask contexts.

Findings

Reconstruction of class posteriors from similarity measures is possible.

Constructed classifiers outperform nearest neighbor classifiers.

Identified problem classes where similarity-based methods are Bayes optimal.

Abstract

For a classification problem described by the joint density $P(\omega,x)$, models of $P(\omega\eq\omega'|x,x')$ (the ``Bayesian similarity measure'') have been shown to be an optimal similarity measure for nearest neighbor classification. This paper analyzes demonstrates several additional properties of that conditional distribution. The paper first shows that we can reconstruct, up to class labels, the class posterior distribution $P(\omega|x)$ given $P(\omega\eq\omega'|x,x')$, gives a procedure for recovering the class labels, and gives an asymptotically Bayes-optimal classification procedure. It also shows, given such an optimal similarity measure, how to construct a classifier that outperforms the nearest neighbor classifier and achieves Bayes-optimal classification rates. The paper then analyzes Bayesian similarity in a framework where a classifier faces a number of related classification tasks (multitask learning) and illustrates that reconstruction of the class posterior distribution is not possible in general. Finally, the paper identifies a distinct class of classification problems using $P(\omega\eq\omega'|x,x')$ and shows that using $P(\omega\eq\omega'|x,x')$ to solve those problems is the Bayes optimal solution.