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Maximum likelihood estimation for the $λ$-exponential family

Xiwei Tian, Ting-Kam Leonard Wong, Jiaowen Yang, Jun Zhang

TL;DR

This paper extends the exponential family to the $\lambda$-exponential family, motivated by optimal transport, and addresses maximum likelihood estimation under i.i.d. sampling where the dual structure is governed by $\lambda$-duality. It develops a fixed-point algorithm in the dual space, deriving a dual gradient and a weight-based update that yields a monotone increase in the log-likelihood for $\lambda<0$. The authors illustrate the method on the $q$-Gaussian distribution and the Dirichlet perturbation, and show that Dirichlet updates can be made independent of $\lambda$, highlighting a practical computational strategy. The results provide a concrete, convergent approach to MLE in generalized exponential families with constant curvature, linking information geometry and optimal transport to scalable inference.

Abstract

The $λ$-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.

Maximum likelihood estimation for the $λ$-exponential family

TL;DR

This paper extends the exponential family to the -exponential family, motivated by optimal transport, and addresses maximum likelihood estimation under i.i.d. sampling where the dual structure is governed by -duality. It develops a fixed-point algorithm in the dual space, deriving a dual gradient and a weight-based update that yields a monotone increase in the log-likelihood for . The authors illustrate the method on the -Gaussian distribution and the Dirichlet perturbation, and show that Dirichlet updates can be made independent of , highlighting a practical computational strategy. The results provide a concrete, convergent approach to MLE in generalized exponential families with constant curvature, linking information geometry and optimal transport to scalable inference.

Abstract

The -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 -Gaussian distribution and the Dirichlet perturbation.
Paper Structure (2 sections, 1 theorem, 13 equations, 1 figure, 1 algorithm)

This paper contains 2 sections, 1 theorem, 13 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. MLE via fixed-point iteration

Key Result

proposition thmcounterproposition

Suppose Assumption ass:assumption holds. If $\hat{\theta}$ is an MLE, i.e. it maximizes $\ell(\theta)$, then $\hat{\eta} := \nabla^{(\lambda)} \varphi_{\lambda}(\hat{\theta})$ satisfies the relation where the weights $w_i$ are defined by

Figures (1)

  • Figure 1: Two trajectories $(\eta(k))$ of Algorithm \ref{['alg:fixed.point']} for the $q$-Gaussian distribution in Example \ref{['eg:q.Gaussian']}. The solid curve shows the graph of $T(\eta)$ on $\Xi = (-\infty, 0)$ defined by \ref{['eqn:iteration']} for the given data-set. The dashed line is the graph of the identity map. The fixed point (indicated by the cross) corresponds to the MLE.

Theorems & Definitions (3)

  • definition thmcounterdefinition: $\lambda$-exponential family
  • proposition thmcounterproposition: First order condition
  • proof