Adaptive Bayesian density estimation using Pitman-Yor or normalized\n inverse-Gaussian process kernel mixtures

2012/10/30 by Catia Scricciolo, Scricciolo, Catia
Biochemistry, Genetics and Molecular Biology · Computer Science · #Bayesian Methods and Mixture Models #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Spectroscopy Techniques in Biomedical and Chemical Research #Statistics Theory (math.ST) #Target Tracking and Data Fusion in Sensor Networks

paper · pdf · doi:10.48550/arxiv.1210.8094

openalex publication_date 2012/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider Bayesian nonparametric density estimation using a Pitman-Yor or a\nnormalized inverse-Gaussian process kernel mixture as the prior distribution\nfor a density. The procedure is studied from a frequentist perspective. Using\nthe stick-breaking representation of the Pitman-Yor process or the expression\nof the finite-dimensional distributions for the normalized-inverse Gaussian\nprocess, we prove that, when the data are replicates from an infinitely smooth\ndensity, the posterior distribution concentrates on any shrinking L^p-norm\nball, 1\≤ p\≤\∞, around the sampling density at a \nearly\nparametric rate, up to a logarithmic factor. The resulting hierarchical\nBayesian procedure, with a fixed prior, is thus shown to be adaptive to the\ninfinite degree of smoothness of the sampling density.\n

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