Duality for real and multivariate exponential families

Abstract

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

25 pages