Duality for real and multivariate exponential families

TL;DR

This paper explores the concept of duality in natural exponential families generated by measures on ^n, focusing on the conditions for the existence of dual measures and their relation to variance functions and Laplace transforms.

Contribution

It introduces a formal framework for duality in exponential families, including the notion of dual measures and their properties, extending the concept to translated families.

Findings

Dual measures exist under specific conditions related to the variance function.

A key property is the inverse relationship between the second derivative of the Laplace transform and the variance function.

The paper extends duality concepts to translated exponential families (TF).

Abstract

Consider a measure $\mu$ on $\R^n$ generating a natural exponential family $F(\mu)$ with variance function $V_{F(\mu)}(m)$ and Laplace transform $$ \exp(\ell_{\mu}(s))=\int_{\R^n} \exp(-\<s,x\>)\mu(dx).$$ A dual measure $\mu^*$ satisfies $-\ell'_{\mu^*}(-\ell'_{\mu}(s))=s.$ Such a dual measure does not always exist. One important property is $\ell"_{\mu^*}(m)=(V_{F(\mu)}(m))^{-1},$ leading to the notion of duality among exponential families (or rather among the extended notion of T exponential families $T\hskip-2pt F$ obtained by considering all translations of a given exponential family $F$).