  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.
