2013/01/31 by XuanLong Nguyen
Computer Science · Mathematics · #Applied mathematics #Artificial intelligence #Base (topology) #Bayes' theorem #Bayesian Methods and Mixture Models #Bayesian probability #Computer science #Data mining #Dirichlet distribution #Dirichlet process #Dirichlet's principle #Generalized Dirichlet distribution #Hierarchical Dirichlet process #Latent Dirichlet allocation #Mathematical analysis #Mathematics #Measure (data warehouse) #Nonparametric statistics #Parametric statistics #Probability measure #Statistical Methods and Inference #Statistics #Stochastic processes and statistical mechanics #Topic model #cs.LG #math.PR #math.ST #stat.TH
paper · pdf · doi:10.3150/15-bej703
published as Bernoulli 2016, Vol. 22, No. 3, 1535-1571 · Published at http://dx.doi.org/10.3150/15-BEJ703 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
openalex publication_date 2016/03/16 · arxiv created 2016/03/24 · arxiv updated 2016/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
This paper studies posterior concentration behavior of the base probability measure of a Dirichlet measure, given observations associated with the sampled Dirichlet processes, as the number of observations tends to infinity. The base measure itself is endowed with another Dirichlet prior, a construction known as the hierarchical Dirichlet processes (Teh et al. [J. Amer. Statist. Assoc. 101 (2006) 1566–1581]). Convergence rates are established in transportation distances (i.e., Wasserstein metrics) under various conditions on the geometry of the support of the true base measure. As a consequence of the theory, we demonstrate the benefit of “borrowing strength” in the inference of multiple groups of data – a powerful insight often invoked to motivate hierarchical modeling. In certain settings, the gain in efficiency due to the latent hierarchy can be dramatic, improving from a standard nonparametric rate to a parametric rate of convergence. Tools developed include transportation distances for nonparametric Bayesian hierarchies of random measures, the existence of tests for Dirichlet measures, and geometric properties of the support of Dirichlet measures.