Dependent mixtures of geometric weights priors

Abstract

A new approach to the joint estimation of partially exchangeable observations is presented. This is achieved by constructing a model with pairwise dependence between random density functions, each of which is modeled as a mixture of geometric stick breaking processes. The claim is that mixture modeling with Pairwise Dependent Geometric Stick Breaking Process (PDGSBP) priors is sufficient for prediction and estimation purposes; that is, making the weights more exotic does not actually enlarge the support of the prior. Moreover, the corresponding Gibbs sampler for estimation is faster and easier to implement than the Dirichlet Process counterpart.

On the inferential implications of decreasing weight structures in mixture models

“…The corresponding mixture model represents a simpler yet appealing alternative to models like those based on the stickbreaking representation. Indeed, the geometric process has been successfully applied in different problems; besides density estimation [Fuentes-García et al, 2010], it has been used in regression [Fuentes-García et al, 2009], dependent models [Mena et al, 2011, Hatjispyros et al, 2018, classification [Gutiérrez et al, 2014], and others. Furthermore, having the weights decreasingly ordered alleviates the label switching effect in the implementation of the posterior sampler and allows for a better interpretation of the weight of each component within the whole sample, thus improving the identifiability of the mixture model [Mena and Walker, 2015].…”

Beta-Binomial stick-breaking non-parametric prior

“…As mentioned above, the ordering of the weights, or lack of it, is of high relevance when using Bayesian nonparametric priors for density estimation and/or clustering. The dependence on only one random variable makes the Geometric process an attractive choice from a numerical point of view, and also makes it quite simple to generalize to non-exchangeable settings 2011;Hatjispyros et al;2018). Furthermore, as shown by Bissiri and Ongaro (2014), both the Dirichlet and the Geometric processes have full support.…”

Bayesian nonparametrics and applications

“…The z ji are independent and identically distributed random variables with density function f j for which we do not assume that they belong to a particular parametric family of densities. Instead we take f j to be nonparametric densities based on the PDGSBP mixture model (Hatjispyros et al, 2017a). The PDGSBP mixture implies the following hierarchical model for the errors.…”

“…In this chapter we focus on the construction of Pairwise Dependent Geometric Stick Breaking Processes (PDGSBP), a dependent Bayesian nonparametric prior for partially exchangeable observations based on the GSB process (Hatjispyros et al, 2017a).…”

“…The method of reconstruction is based on the Pairwise Dependent Geometric Stick Breaking Processes (PDGSBP) mixture priors (Hatjispyros et al, 2017a) already described in Chapter 4.…”