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Camerlenghi, Federico; Dunson, David B.; Lijoi, Antonio; Prünster, Igor; Rodríguez, Abel
[Beraha, Mario; Guglielmi, Alessandra; Liu, Vera; Müller, Peter; Quintana, Fernando A.; Jara, Alejandro; Ma, Li; Robert, Christian P.; Leisen, Fabrizio; Riva Palacio, Alan; Aliverti, Emanuele; Paganin, Sally; Rigon, Tommaso; Russo, Massimiliano]

Latent nested nonparametric priors (with discussion). (English) Zbl 1436.62108

Bayesian Anal. 14, No. 4, 1303-1356 (2019).
Summary: Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalizing to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop a Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by-product. The results and their inferential implications are showcased on synthetic and real data.

Cited in 12 Documents

MSC:

62G05 Nonparametric estimation
60G57 Random measures
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62P10 Applications of statistics to biology and medical sciences; meta analysis

Keywords:

Bayesian nonparametrics; completely random measures; dependent nonparametric priors; heterogeneity; mixture models; nested processes

Software:

shallot
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\textit{F. Camerlenghi} et al., Bayesian Anal. 14, No. 4, 1303--1356 (2019; Zbl 1436.62108)
Full Text: DOI arXiv Euclid

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