Bayesian Inference via Classes of Normalized Random Measures

Abstract

One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. Here we exploit theoretical properties of Poisson random measures in order to provide a comprehensive Bayesian analysis of random probabilities which are obtained by an appropriate normalization. Specifically we achieve explicit and tractable forms of the posterior and the marginal distributions, including an explicit and easily used description of generalizations of the important Blackwell-MacQueen Pólya urn distribution. Such simplifications are achieved by the use of a latent variable which admits quite interesting interpretations which allow to gain a better understanding of the behaviour of these random probability measures. It is noteworthy that these models are generalizations of models considered by Kingman (1975) in a non-Bayesian context. Such models are known to play a significant role in a variety of applications including genetics, physics, and work involving random mappings and assemblies. Hence our analysis is of utility in those contexts as well. We also show how our results may be applied to Bayesian mixture models and describe computational schemes which are generalizations of known efficient methods for the case of the Dirichlet process. We illustrate new examples of processes which can play the role of priors for Bayesian nonparametric inference and finally point out some interesting connections with the theory of generalized gamma convolutions initiated by Thorin and further developed by Bondesson.

On Bayesian nonparametric modelling of two correlated distributions

“…So, we can construct correlated distributions with DP marginals with parameters M and H by taking weighted sums of independent DPs with the same base distribution H. This idea could be extended to larger numbers of distributions or any process constructed by normalising a random measure with independent increments (James et al, 2005), as discussed by Griffin et al (2010). However, the model with Dirichlet process marginals is the only one where the weights are independent of the component random distributions.…”

The Ornstein–Uhlenbeck Dirichlet process and other time-varying processes for Bayesian nonparametric inference

“…3 The OUNRM processes The Ferguson and Klass (1972) representation of the BNS OU Lévy process and James et al (2005)'s representation of the NRM process are both expressed through functions of Poisson processes. The OUNRM process combines these two ideas to give a Poisson processbased definition.…”

“…I will concentrate on the OUDP case although the methods can be simply extended to other OUNRM processes. For example, updating parameters connected to G 0 could be implemented using the methodology of James et al (2005). In general, the number of subsequent jumps will be almost surely infinite in any region and a method for truncation is described in Gander and Stephens (2006).…”

A Monte Carlo Markov chain algorithm for a class of mixture time series models

“…Ferguson's (1973) Dirichlet process is a well-known random probability measure that has been adopted in Bayesian nonparametric research for the last three decades. Lately, the Kingman-Pitman-Yor Poisson-Dirichlet process (Kingman 1975;Pitman and Yor 1997) and James' normalized generalized Gamma process (James 2002(James , 2005 have attracted attention in the field of Bayesian nonparametric statistics (Ho 2006b;James 2001, 2004;James et al 2008;Lau and Green 2007;Lijoi et al 2005Lijoi et al , 2007. The Dirichlet process we consider is almost surely discrete and can be represented as a weighted sum of the Dirac delta functions at the independent draws of G 0 ,…”

“…This section discusses possible extensions to other time series models by adopting more general random measures, for instance, the two-parameter Poisson-Dirichlet processes in Pitman and Yor (1997) (see also Pitman 1995, 1996, 2003and Kingman 1975, 1993 and the normalized completely random measures in James et al (2008) and James (2002James ( , 2005) (see also Lijoi et al 2005, for the special case of the normalized inverse Gaussian processes). The model in (1) can be generalized to Fig.…”