Adaptive Bayesian multivariate density estimation with Dirichlet mixtures

Abstract

We show that rate-adaptive multivariate density estimation can be performed using Bayesian methods based on Dirichlet mixtures of normal kernels with a prior distribution on the kernel's covariance matrix parameter. We derive sufficient conditions on the prior specification that guarantee convergence to a true density at a rate that is optimal minimax for the smoothness class W. SHEN, S.T. TOKDAR AND S. GHOSAL to which the true density belongs. No prior knowledge of smoothness is assumed. The sufficient conditions are shown to hold for the Dirichlet location mixture of normals prior with a Gaussian base measure and an inverse-Wishart prior on the covariance matrix parameter. Locally Hölder smoothness classes and their anisotropic extensions are considered. Our study involves several technical novelties, including sharp approximation of finitely differentiable multivariate densities by normal mixtures and a new sieve on the space of such densities.

Adaptive Bayesian Procedures Using Random Series Priors

“…Applying Theorem 2 with N n D .n= log n/ ˛ =.2˛ Cs/ , J n D n s=.2˛ Cs/ .log n/ .2˛ /=.2˛ Cs/ t2 , M n D n 1=t3 , a 4 D 1=2 and r D 1, the posterior distribution contracts at p 0 with respect to the Hellinger distance at the rate n D n ˛ =.2˛ Cs/ .log n/˛ =.2˛ Cs/C.1 t2/=2 . Essentially, the same rate is also obtained in Shen et al (2013) (with a different logarithmic factor) using a Dirichlet mixture of normal prior.…”

“…Essentially, the same rate is also obtained in Shen et al . () (with a different logarithmic factor) using a Dirichlet mixture of normal prior.…”

Bayesian Repulsive Gaussian Mixture Model

“…We will also discuss possible extensions to the case where p λ has full support on (0, +∞) later in this section. Condition B5 is the requirement for the tail behavior of the prior on K. Similar assumption on the tail behavior of the prior on K is adopted in Kruijer et al (2010) and Shen et al (2013) for finite mixture models. As a useful example, we show that the commonly used zero-truncated Poisson prior on K satisfies condition B5.…”

Anisotropic function estimation using multi-bandwidth Gaussian processes

“…There is a growing literature on Bayesian adaptation over the last decade. Previous works include Belitser and Ghosal (2003); de Jonge and van Zanten (2010); Ghosal, Lember and van der Vaart (2003); Ghosal, Lember and Van Der Vaart (2008); Huang (2004); Kruijer, Rousseau and van der Vaart (2010); Rousseau (2010); Scricciolo (2006); Shen and Ghosal (2011).…”