On Tsallis Entropy Bias and Generalized Maximum Entropy Models

TL;DR

This paper introduces a generalized Maxent model based on Tsallis entropy that corrects for entropy bias, improving density estimation by reducing overfitting and underfitting through a novel entropy bias correction method.

Contribution

It proposes a new Tsallis entropy bias correction framework for Maxent models, providing a closed-form formula and connecting it with Lidstone estimators for better density estimation.

Findings

TEBC Maxent outperforms existing density estimation methods.

The TEB-Lidstone estimator effectively corrects probability estimates.

Empirical results demonstrate improved accuracy and robustness.

Abstract

In density estimation task, maximum entropy model (Maxent) can effectively use reliable prior information via certain constraints, i.e., linear constraints without empirical parameters. However, reliable prior information is often insufficient, and the selection of uncertain constraints becomes necessary but poses considerable implementation complexity. Improper setting of uncertain constraints can result in overfitting or underfitting. To solve this problem, a generalization of Maxent, under Tsallis entropy framework, is proposed. The proposed method introduces a convex quadratic constraint for the correction of (expected) Tsallis entropy bias (TEB). Specifically, we demonstrate that the expected Tsallis entropy of sampling distributions is smaller than the Tsallis entropy of the underlying real distribution. This expected entropy reduction is exactly the (expected) TEB, which can be expressed by a closed-form formula and act as a consistent and unbiased correction. TEB indicates that the entropy of a specific sampling distribution should be increased accordingly. This entails a quantitative re-interpretation of the Maxent principle. By compensating TEB and meanwhile forcing the resulting distribution to be close to the sampling distribution, our generalized TEBC Maxent can be expected to alleviate the overfitting and underfitting. We also present a connection between TEB and Lidstone estimator. As a result, TEB-Lidstone estimator is developed by analytically identifying the rate of probability correction in Lidstone. Extensive empirical evaluation shows promising performance of both TEBC Maxent and TEB-Lidstone in comparison with various state-of-the-art density estimation methods.