## Bernoulli

• Bernoulli
• Volume 23, Number 1 (2017), 379-404.

### Posterior asymptotics of nonparametric location-scale mixtures for multivariate density estimation

#### Abstract

Density estimation represents one of the most successful applications of Bayesian nonparametrics. In particular, Dirichlet process mixtures of normals are the gold standard for density estimation and their asymptotic properties have been studied extensively, especially in the univariate case. However, a gap between practitioners and the current theoretical literature is present. So far, posterior asymptotic results in the multivariate case are available only for location mixtures of Gaussian kernels with independent prior on the common covariance matrix, while in practice as well as from a conceptual point of view a location-scale mixture is often preferable. In this paper, we address posterior consistency for such general mixture models by adapting a convergence rate result which combines the usual low-entropy, high-mass sieve approach with a suitable summability condition. Specifically, we establish consistency for Dirichlet process mixtures of Gaussian kernels with various prior specifications on the covariance matrix. Posterior convergence rates are also discussed.

#### Article information

Source
Bernoulli Volume 23, Number 1 (2017), 379-404.

Dates
Revised: June 2015
First available in Project Euclid: 27 September 2016

https://projecteuclid.org/euclid.bj/1475001358

Digital Object Identifier
doi:10.3150/15-BEJ746

Mathematical Reviews number (MathSciNet)
MR3556776

Zentralblatt MATH identifier
1377.62106

#### Citation

Canale, Antonio; De Blasi, Pierpaolo. Posterior asymptotics of nonparametric location-scale mixtures for multivariate density estimation. Bernoulli 23 (2017), no. 1, 379--404. doi:10.3150/15-BEJ746. https://projecteuclid.org/euclid.bj/1475001358

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