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We extend Fanos inequality, which controls the average probability of events in terms of the average of some $f$--divergences, to work with arbitrary events (not necessarily forming a partition) and even with arbitrary $[0,1]$--valued random variables, possibly in continuously infinite number. We provide two applications of these extensions, in which the consideration of random variables is particularly handy: we offer new and elegant proofs for existing lower bounds, on Bayesian posterior concentration (minimax or distribution-dependent) rates and on the regret in non-stochastic sequential learning.
The variance and the entropy power of a continuous random variable are bounded from below by the reciprocal of its Fisher information through the Cram{e}r-Rao bound and the Stams inequality respectively. In this note, we introduce the Fisher informat
We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Speci
In this paper we prove the concavity of the $k$-trace functions, $Amapsto (text{Tr}_k[exp(H+ln A)])^{1/k}$, on the convex cone of all positive definite matrices. $text{Tr}_k[A]$ denotes the $k_{mathrm{th}}$ elementary symmetric polynomial of the eige
This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the ErdH{o}s-Renyi m
We prove large (and moderate) deviations for a class of linear combinations of spacings generated by i.i.d. exponentially distributed random variables. We allow a wide class of coefficients which can be expressed in terms of continuous functions defi