2014/12/31 by Daniel M. Roy, Roy, Daniel M.
Computer Science · Economics, Econometrics and Finance · Mathematics · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Financial Risk and Volatility Modeling #Machine Learning (stat.ML) #Probability (math.PR) #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1501.00208
openalex publication_date 2014/12/31 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
We describe the combinatorial stochastic process underlying a sequence of\nconditionally independent Bernoulli processes with a shared beta process hazard\nmeasure. As shown by Thibaux and Jordan [TJ07], in the special case when the\nunderlying beta process has a constant concentration function and a finite and\nnonatomic mean, the combinatorial structure is that of the Indian buffet\nprocess (IBP) introduced by Griffiths and Ghahramani [GG05]. By reinterpreting\nthe beta process introduced by Hjort [Hjo90] as a measurable family of\nDirichlet processes, we obtain a simple predictive rule for the general case,\nwhich can be thought of as a continuum of Blackwell-MacQueen urn schemes (or\nequivalently, one-parameter Hoppe urn schemes). The corresponding measurable\nfamily of Perman-Pitman-Yor processes leads to a continuum of two-parameter\nHoppe urn schemes, whose ordinary component is the three-parameter IBP\nintroduced by Teh and G "or "ur [TG09], which exhibits power-law behavior, as\nfurther studied by Broderick, Jordan, and Pitman [BJP12]. The idea extends to\narbitrary measurable families of exchangeable partition probability functions\nand gives rise to generalizations of the beta process with matching buffet\nprocesses. Finally, in the same way that hierarchies of Dirichlet processes\nwere given Chinese restaurant franchise representations by Teh, Jordan, Beal,\nand Blei [Teh+06], one can construct representations of sequences of Bernoulli\nprocesses directed by hierarchies of beta processes (and their generalizations)\nusing the stochastic process we uncover.\n