Abstract
The predictive probabilities of the hierarchical Pitman-Yor process are approximated through Monte Carlo algorithms that exploits the Chinese Restaurant Franchise (CRF) representation. However, in order to simulate the posterior distribution of the hierarchical Pitman-Yor process, a set of auxiliary variables representing the arrangement of customers in tables of the CRF must be sampled through Markov chain Monte Carlo. This paper develops a perfect sampler for these latent variables employing ideas from the Propp-Wilson algorithm and evaluates its average running time by extensive simulations. The simulations reveal a significant dependence of running time on the parameters of the model, which exhibits sharp transitions. The algorithm is compared to simpler Gibbs sampling procedures, as well as a procedure for unbiased Monte Carlo estimation proposed by Glynn and Rhee. We illustrate its use with an example in microbial genomics studies.
Highlights
The hierarchical Pitman–Yor process was introduced in Teh et al (2006) and Teh (2006) as a nonparametric prior model for a collection of discrete distributions with heavy tails
In Bayesian nonparametrics, theoretical developments and applications of the hierarchical Pitman–Yor process have been considered in language modeling (Teh, 2006; Huang and Renals, 2007; Wood et al, 2009), infinite hidden Markov modeling (Beal et al, 2002; Van Gael et al, 2008; Blunsom and Cohn, 2011), species sampling with multiple populations (Battiston et al, 2018; Camerlenghi et al, 2019; Bassetti et al, 2020; Camerlenghi et al, 2019), clustering (Argiento et al, 2020), graphical modeling
The predictive probabilities induced by the hierarchical Pitman–Yor process can be described by means of the Chinese Restaurant Franchise (CRF), which extends the Chinese Restaurant Process (CRP)
Summary
The hierarchical Pitman–Yor process was introduced in Teh et al (2006) and Teh (2006) as a nonparametric prior model for a collection of discrete distributions with heavy tails. See Teh and Jordan (2010) for a review on hierarchical nonparametric priors. We discuss: i) a novel conditional Gibbs sampling; ii) an application of coupling from the past for perfect sampling; iii) the application of a more recent framework studied by Glynn and Rhee (2014) with the aim of deriving unbiased posterior estimates from Markov chain sampling. The latter approach can be viewed as an intermediate solution between Gibbs sampling and perfect simulations
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