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 $F$).
Categories: math.PR