On the Pitman–Yor process with spike and slab base measure

Abstract

Summary For the most popular discrete nonparametric models, beyond the Dirichlet process, the prior guess at the shape of the data-generating distribution, also known as the base measure, is assumed to be diffuse. Such a specification greatly simplifies the derivation of analytical results, allowing for a straightforward implementation of Bayesian nonparametric inferential procedures. However, in several applied problems the available prior information leads naturally to the incorporation of an atom into the base measure, and then the Dirichlet process is essentially the only tractable choice for the prior. In this paper we fill this gap by considering the Pitman–Yor process with an atom in its base measure. We derive computable expressions for the distribution of the induced random partitions and for the predictive distributions. These findings allow us to devise an effective generalized Pólya urn Gibbs sampler. Applications to density estimation, clustering and curve estimation, with both simulated and real data, serve as an illustration of our results and allow comparisons with existing methodology. In particular, we tackle a functional data analysis problem concerning basal body temperature curves.

Hierarchical Species Sampling Models

“…atomic). Moreover, properties of these sequences are also relevant for studying non-parametric prior distributions with mixed base measure, such as the Pitman-Yor process with spike-and-slab base measure introduced in Canale et al (2017). Given a random partition Π, let C j (Π) be the random index of the block containing j, that is…”

Stochastic Approximations to the Pitman–Yor Process

“…It originates from the work of Perman et al (1992), further investigated in Pitman (1995); Pitman and Yor (1997), and its use in nonparametric inference was initiated by Ishwaran and James (2001). Thanks to its analytical tractability and flexibility, it has found applications in a variety of inferential problems which include species sampling (Lijoi et al, 2007;Favaro et al, 2009;Navarrete et al, 2008), survival analysis and gene networks (Jara et al, 2010;Ni et al, 2018), linguistics and image segmentation (Teh, 2006;Sudderth and Jordan, 2009), curve estimation (Canale et al, 2017) and time-series and econometrics (Caron et al, 2017;Bassetti et al, 2014). The Pitman-Yor process is a discrete probability measure…”

Asymptotic number of clusters for species sampling sequences with non-diffuse base measure

“…Our approach is different from the one used in Sangalli (2006) and Canale et al (2017), which is based on specific properties of nomalized random measures. Using combinatorial arguments developed in Pitman (2006), we are able to consider more general species sampling sequences and study their asymptotic properties.…”

“…Dunson et al (2008); Kim et al (2009); Suarez and Ghosal (2016); Cui and Cui (2012); Barcella et al (2016). Spike and slab base measures have also been considered for a Pitman Yor process in Canale et al (2017), where computable expressions for the distribution of the random partitions induced by such a process are derived and used for predictive inference.…”