Maximum Entropy competes with Maximum Likelihood
This work provides a systematic evaluation of MAXENT's applicability and limitations for researchers and practitioners using probabilistic inference, particularly in sparse data scenarios.
The paper investigates the validity limits of the Maximum Entropy (MAXENT) method within a Bayesian decision theory framework, comparing it against regularized Maximum Likelihood (ML) and Bayesian estimators. It finds that MAXENT is effective in sparse data regimes and can outperform optimally regularized ML if there are prior rank correlations between the estimated quantity and its probabilities.
Maximum entropy (MAXENT) method has a large number of applications in theoretical and applied machine learning, since it provides a convenient non-parametric tool for estimating unknown probabilities. The method is a major contribution of statistical physics to probabilistic inference. However, a systematic approach towards its validity limits is currently missing. Here we study MAXENT in a Bayesian decision theory set-up, i.e. assuming that there exists a well-defined prior Dirichlet density for unknown probabilities, and that the average Kullback-Leibler (KL) distance can be employed for deciding on the quality and applicability of various estimators. These allow to evaluate the relevance of various MAXENT constraints, check its general applicability, and compare MAXENT with estimators having various degrees of dependence on the prior, viz. the regularized maximum likelihood (ML) and the Bayesian estimators. We show that MAXENT applies in sparse data regimes, but needs specific types of prior information. In particular, MAXENT can outperform the optimally regularized ML provided that there are prior rank correlations between the estimated random quantity and its probabilities.