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The combinatorial structure of beta negative binomial processes. (English) Zbl 1358.60069

This paper treats a characterization for the combinatorial structure of conditionally i.i.d sequences of negative binomial processes \(X_j\) \(\sim\) \(\mathrm{NBP}(r, B_0)\) with a common beta process base measure \(B_0\) on \(( \Omega, {\mathcal A})\), where \(r > 0\) is a parameter. In a Bayesian non-parametric applicatory framework, such processes have served as models for latent multisets of features underlying certain data. Analogously, random subsets arise from conditionally i.i.d. sequences of Bernoulli processes with a common beta process base measure. In this case, the combinatorial structure is described by the Indian buffet process (IBP). The principal feature of the proof consists in the description of the key Markov kernels needed to use a negative binomial Indian buffet process (NB-IBP) representation in a Markov chain Monte Carlo algorithm targeting a posterior distribution. For other related works, see, e.g., [T. Broderick et al., “Combinatorial clustering and the beta-negative binomial process”, IEEE Trans. Pattern Anal. Mach. Intell. 37, No. 2, 290–306, (2014; doi:10.1109/TPAMI.2014.2318721); N. L. Hjort, Ann. Stat. 18, No. 3, 1259–1294 (1990; Zbl 0711.62033); G. Gregoire, Stochastic Processes Appl. 16, 179–188 (1984; Zbl 0522.60048)].

MSC:

60G99 Stochastic processes
60G09 Exchangeability for stochastic processes
60G57 Random measures
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J22 Computational methods in Markov chains
65C05 Monte Carlo methods
65C40 Numerical analysis or methods applied to Markov chains
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References:

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