Beyond scalar quasi-arithmetic means: Quasi-arithmetic averages and quasi-arithmetic mixtures in information geometry

TL;DR

This paper extends quasi-arithmetic means to a geometric setting using Legendre functions, exploring their properties, duality, and applications in statistical models within information geometry.

Contribution

It introduces a generalization of quasi-arithmetic means through Legendre functions and analyzes their invariance, duality, and applications in statistical models in information geometry.

Findings

Generalization of quasi-arithmetic means via Legendre functions.

Duality of quasi-arithmetic averages through convex conjugates.

Applications to statistical models closed under quasi-arithmetic mixtures.

Abstract

We generalize quasi-arithmetic means beyond scalars by considering the gradient map of a Legendre type real-valued function. The gradient map of a Legendre type function is proven strictly comonotone with a global inverse. It thus yields a generalization of strictly mononotone and differentiable functions generating scalar quasi-arithmetic means. Furthermore, the Legendre transformation gives rise to pairs of dual quasi-arithmetic averages via the convex duality. We study the invariance and equivariance properties under affine transformations of quasi-arithmetic averages via the lens of dually flat spaces of information geometry. We show how these quasi-arithmetic averages are used to express points on dual geodesics and sided barycenters in the dual affine coordinate systems. We then consider quasi-arithmetic mixtures and describe several parametric and non-parametric statistical models which are closed under the quasi-arithmetic mixture operation.