Entropy and the fourth moment phenomenon

Abstract

We develop a new method for bounding the relative entropy of a random vector in terms of its Stein factors. Our approach is based on a novel representation for the score function of smoothly perturbed random variables, as well as on the de Bruijn's formula of information theory. When applied to sequences of functionals of a general Gaussian field, our results can be combined with the Carbery-Wright inequality in order to yield multidimensional entropic rates of convergence that coincide, up to a logarithmic factor, with those achievable in smooth distances (such as the 1-Wasserstein distance). In particular, our findings settle the open problem of proving a quantitative version of the multidimensional fourth moment theorem for random vectors having chaotic components, with explicit rates of convergence in total variation that are independent of the order of the associated Wiener chaoses. The results proved in the present paper are outside the scope of other existing techniques, such as for instance the multidimensional Stein's method for normal approximations.

Distances between nested densities and a measure of the impact of the prior in Bayesian statistics

“…The Stein kernels for a wide variety of classical distributions (all members of the Pearson family, as it turns out) bear agreeable expressions; see [8, Table 1], [20,21] or the forthcoming [7] for illustrations.…”

Quantitative clts on a gaussian space: a survey of recent developments

“…One should notice, however, that (similarly to equation (4.42)) the rates of convergence in the total variation distance implied by [28] heavily depend on the order of the chaoses -whereas it is natural to conjecture that they should be the same as those deduced in [32,Chapter 6] for the multidimensional Wasserstein distance. This difficulty has been partially solved in the paper [37], where the main estimates proved in [28] are combined with tools from information theory.…”

“…As a first step in this direction, we now present a statement (Theorem 3.7) connecting Stein factors to the shape of the density of a random variable F ∈ D 1,2 having an absolutely continuous distribution. A proof of the first part (which is originally due to Nourdin and Viens) can be found in [32, Theorem 10.1.1], while the second part is a new powerful representation of relative entropy, that is the starting point of the paper [37].…”

“…The monograph [32] provides a quite detailed introduction to the main object of the present paper, with specific emphasis on the so-called 'fourth moment theorems' (see [42]), as well as on the interaction between Malliavin calculus, Stein's method and Poincaré-type estimates. Since the publication of [32] some new techniques have been introduced (see [28,37,38]) -in particular related to Carbery-Wright inequalities (see [6]) and tools from information theory. One of my implicit aims is to provide a quick introduction to these recent developments.…”

Recent advances in various fields of numerical probability

“…This allows to derive integration by parts formulas and to obtain convenient estimates for the weights which appear in these formulas. It is worth mentioning that a variant of the Malliavin calculus based on a similar splitting method has already been used by Nourdin and Poly [59] (see also [60] and [54]). They use the so-called Γ-calculus (see e.g.…”