Duality for real and multivariate exponential families

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 $F$).