Bayesian inference with dependent normalized completely random measures

By Antonio Lijoi, Bernardo Nipoti, Igor Prünster. 2014

94 citations recorded by the source corpus.

The proposal and study of dependent prior processes has been a major research focus in the recent Bayesian nonparametric literature. In this paper, we introduce a flexible class of dependent nonparametric priors, investigate their properties and derive a suitable sampling scheme which allows their concrete implementation. The proposed class is obtained by normalizing dependent completely random measures, where the dependence arises by virtue of a suitable construction of the Poisson random measures underlying the completely random measures. We first provide general distributional results for the whole class of dependent completely random measures and then we specialize them to two specific priors, which represent the natural candidates for concrete implementation due to their analytic tractability: the bivariate Dirichlet and normalized $\sigma$-stable processes. Our analytical results, and in particular the partially exchangeable partition probability function, form also the basis for the determination of a Markov Chain Monte Carlo algorithm for drawing posterior inferences, which reduces to the well-known Blackwell--MacQueen P\'{o}lya urn scheme in the univariate case. Such an algorithm can be used for density estimation and for analyzing the clustering structure of the data and is illustrated through a real two-sample dataset example.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f449c76838cca52288a3dd458224af088b7f4369

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Citations in the corpus

The Dependent Dirichlet Process and Related Models (2020, methodology): “We refer the reader to Lijoi and Prünster (2009) for an exhaustive summary of existing results concerning distributional properties of functional of single and discrete random probability measures. In the context of a single probability measure, Kessler, Hoff and Dunson (2015) proposed a clever cons”
Compound random measures and their use in Bayesian non‐parametrics (2014, background): “3 Compound Random Measures In this section, we will define a general class of vectors of NRMI that incorporates many recently proposed priors built using normalization, see for instance [28], [29], [49], [18] and [32].”
Comparing distributions by using dependent normalized random‐measure mixtures (2013, background): “Lijoi et al. (2011) derived the same expression (but with rather different notation) for the case of two groups.”