Duality for real and multivariate exponential families

Abstract

Consider a measure μ on Rn generating a natural exponential family F(μ) with variance function VF(μ)(m) and Laplace transform exp(ℓμ(s))=∫Rnexp(−〈x,s〉μ(dx)).A dual measure μ∗ satisfies −ℓμ∗′(−ℓμ′(s))=s. Such a dual measure does not always exist. One important property is ℓμ∗′′(m)=(VF(μ)(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).