Some aspects of symmetric Gamma process mixtures

Abstract

In this article, we present some specific aspects of symmetric Gamma process mixtures for use in regression models. We propose a new Gibbs sampler for simulating the posterior and we establish adaptive posterior rates of convergence related to the Gaussian mean regression problem.

Bayesian nonparametric estimation for Quantum Homodyne Tomography

“…Simulation results. We use the algorithm of Naulet and Barat (2015) for simulating samples from posterior distributions of Gamma process mixtures. The base measure α on R 2 × [0, 2π] of the mixing Gamma process is taken as the independent product of a normal distribution on R 2 with covariance matrix diag(1/2, 1/2) and the uniform distribution on [0, 2π].…”

Tails assumptions and posterior concentration rates for mixtures of Gaussians

“…Notice that equation ( 5) forbids the use of the classical inverse-Gamma distribution as prior distribution on σ because of its heavy tail. In fact, it is always possible to weaken equation ( 5) to allow for Inverse-Gamma distribution (see Canale and De Blasi (2013); Naulet and Barat (2015)) but it complicates the proofs with no contribution to the subject of the paper. We found that among the usual distributions the inverse-Gaussian is more suitable for our purpose since it fulfills all the equations (5) to (7), as shown in proposition 1.…”

“…Nonparametric mixture models are highly popular in the Bayesian nonparametric literature, due to both their reknown flexibility and relative easiness of implementation, see Hjort et al (2010) for a review. They have been used in particular for density estimation, clustering and classification and recently nonparametric mixture models have also been proposed in nonlinear regression models, see for instance de Jonge and van Zanten (2010); Wolpert et al (2011); Naulet and Barat (2015).…”

“…We assume that s is known, which is just a matter of convenience for proofs. All the results of the paper can be translated to the case s unknown using the same methodology as Salomond (2013) or Naulet and Barat (2015). Our aim is to study posterior concentration rates in L 2 (Q 0 ) around the true regression function f 0 defined by sequences ǫ n converging to zero with n and such that (4)…”

“…For any finite positive measure α on the measurable space (X, X ), let Π α denote the symmetric Gamma process distribution with parameter α (Wolpert et al, 2011;Naulet and Barat, 2015); that is, an M ∼ Π α is a random signed measure on (X, X ) such that far any disjoints B 1 , . .…”