Duality for real and multivariate exponential families

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

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