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.
Sergio Bacallado and Samuel Power received funding from the Cantab Capital Institute for the Mathematics of Information. Stefano Favaro received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 817257. Stefano Favaro gratefully acknowledge the financial support from the Italian Ministry of Education, University and Research (MIUR), “Dipartimenti di Eccellenza” grant 2018–2022. Lorenzo Trippa has been supported by the Claudia Adams Barr Program in Cancer Research and the NIH grant 1R01LM013352-01A1.
The authors are grateful to an Associate Editor and two anonymous Referees for all their comments, corrections, and suggestions which improved remarkably the paper.
"Perfect Sampling of the Posterior in the Hierarchical Pitman–Yor Process." Bayesian Anal. Advance Publication 1 - 25, 2021. https://doi.org/10.1214/21-BA1269