Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparamatric estimation of the density

By Judith Rousseau. 2010

78 citations recorded by the source corpus.

In this work we investigate the asymptotic properties of nonparametric bayesian mixtures of Betas for estimating a smooth density on [0,1]. We consider a parameterisation of Betas distributions in terms of mean and scale parameters and construct a mixture of these Betas in the mean parameter, while putting a prior on this scaling parameter. We prove that such Bayesian nonparametric models have good frequentist asymptotic properties. We determine the posterior rate of concentration around the true density and prove that it is the minimax rate of concentration when the true density belongs to a Holder class with regularity β, for all positive β, leading to a minimax adaptive estimating procedure of the density. We show that Bayesian kernel estimation is more ﬂexible than the usual frequentist kernel estimation allowing for adaptive rates of convergence, using a simple trick which can be used in many other types of kernel Bayesian approaches.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3e8a2ed091f3df7cf4cef070b9583b14e37b2e3c

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Citations in the corpus

On adaptive posterior concentration rates (2013, background): “f Dirichlet process mixtures or related types of mixtures lead to adaptive posterior concentration rates over collections of H¨older balls of regularity β, up to a lognterm, see for instance Rousseau [24], Kruijer et al. [20], Shen et al. [28] and Scricciolo [27] under the Hellinger or the L1 losses”
Semiparametric Inference in Dynamic Binary Choice Models, Second Version (2014, background): “Mixtures of beta distributions can approximate and consistently estimate large non-parametric classes of densities; see, for example, Rousseau (2010).”
POSTERIOR CONSISTENCY IN CONDITIONAL DENSITY ESTIMATION BY COVARIATE DEPENDENT MIXTURES (2011, background): “For example, Rousseau (2010) and Kruijer, Rousseau, and van der Vaart (2010) show that in univariate density estimation, Bayesian models based on mixtures of distributions automatically adapt to the smoothness of the target density and deliver minimax convergence rates up to a logarithmic factor.”
Adaptive Bayesian density estimation with location-scale mixtures (2010, methodology): “A recent result by Rousseau [26] however does allow for nonnegative approximation of smooth but compactly supported densities by beta-mixtures.”
Posterior Contraction Rates for the Bayesian Approach to Linear Ill-Posed Inverse Problems (2012, unknown): “particular the problem of posterior consistency has attracted a lot of interest in recent years; see for instance [10,24,23,27,26,11,7].”