Section: New Results

Bayesian methods

Participants : Gérard Biau, Vincent Rivoirard.

The ABC method

Approximate Bayesian Computation (ABC for short) is a family of computational techniques which offer an almost automated solution in situations where evaluation of the posterior likelihood is computationally prohibitive, or whenever suitable likelihoods are not available. In the paper [29] Gérard Biau and his coauthors analyze the procedure from the point of view of $k$-nearest neighbor theory and explore the statistical properties of its outputs. They discuss in particular some asymptotic features of the genuine conditional density estimate associated with ABC, which is a new interesting hybrid between a $k$-nearest neighbor and a kernel method.

Semi-parametric version of the Bernstein-von Mises theorem

In [22] , Vincent Rivoirard and Judith Rousseau study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semi-parametric version of the Bernstein-von Mises theorem. The special case of the cumulative distributive function evaluated at a specific point is widely considered. In particular, they show that for infinite dimensional exponential families, under quite general assumptions, the asymptotic posterior distribution of the functional can be either Gaussian or a mixture of Gaussian distributions with different centering points. This illustrates the positive but also the negative phenomena that can occur for the study of Bernstein-von Mises results. In [22] Vincent Rivoirard and Judith Rousseau use convergence rates on Besov spaces established in [23] .