Maximum likelihood estimation for the $\lambda$-exponential family

TL;DR

This paper introduces a fixed point iteration method for maximum likelihood estimation in the $\lambda$-exponential family, a generalized form of the exponential family motivated by optimal transport, and demonstrates its effectiveness with specific distributions.

Contribution

It proposes a novel fixed point algorithm for MLE in the $\lambda$-exponential family and proves its monotonic likelihood property using duality theory.

Findings

Likelihood increases monotonically along the proposed iterations.

Algorithm successfully applied to $q$-Gaussian and Dirichlet perturbation distributions.

Provides a computational approach for the $\lambda$-exponential family.

Abstract

The $\lambda$-exponential family generalizes the standard exponential family via a generalized convex duality motivated by optimal transport. It is the constant-curvature analogue of the exponential family from the information-geometric point of view, but the development of computational methodologies is still in an early stage. In this paper, we propose a fixed point iteration for maximum likelihood estimation under i.i.d.~sampling, and prove using the duality that the likelihood is monotone along the iterations. We illustrate the algorithm with the $q$-Gaussian distribution and the Dirichlet perturbation.