Renyi, Kullback-Leibler, $f$ divergences and exponential families.

par Gérard Letac (IMT)

mardi 27 mai 2025, 11:15 → 12:15 Europe/Paris

Salle K. Johnson (1R3, 1er étage)

 Starting from the observation that the Kullback Leibler divergence of two bimomial laws $P=B(n,p)$ and $P_1=B(n,p_1)$ is a linear function of $n$, we study which divergences have this kind  of behavior when applied to natural exponential families (NEF) and their convolution powers. Conversely we wonder which pairs $(P,P_1)$ are such that $D(P^*||P_1^n)=nD(P||P_1) $ for these divergences $D$. The hoped answer is the fact that $P$ and $P_1$ belong to the same NEF, but this is not true for rather pathologic cases.
This is part of a paper written with Mauro Piccioni 'Exponential families, Renyi divergence and almost sure Cauchy equation' J. Theoret. Probab. 2025.