We consider predictive inference using a class of temporally dependent Dirichlet processes driven by Fleming–Viot diffusions, which have a natural bearing in Bayesian nonparametrics and lend the resulting family of random probability measures to analytical posterior analysis. Formulating the implied statistical model as a hidden Markov model, we fully describe the predictive distribution induced by these Fleming–Viot-driven dependent Dirichlet processes, for a sequence of observations collected at a certain time given another set of draws collected at several previous times. This is identified as a mixture of Pólya urns, whereby the observations can be values from the baseline distribution or copies of previous draws collected at the same time as in the usual Pólya urn, or can be sampled from a random subset of the data collected at previous times. We characterize the time-dependent weights of the mixture which select such subsets and discuss the asymptotic regimes. We describe the induced partition by means of a Chinese restaurant process metaphor with a conveyor belt, whereby new customers who do not sit at an occupied table open a new table by picking a dish either from the baseline distribution or from a time-varying offer available on the conveyor belt. We lay out explicit algorithms for exact and approximate posterior sampling of both observations and partitions, and illustrate our results on predictive problems with synthetic and real data.
The authors are grateful to an Associate Editor and two anonymous referees for carefully reading the manuscript and for providing helpful and constructive comments. The second and third authors are partially supported by the Italian Ministry of Education, University and Research (MIUR) through PRIN 2015SNS29B. The third author is also supported by MIUR through “Dipartimenti di Eccellenza” grant 2018-2022. Helpful discussions with Amil Ayoub are gratefully acknowledged by the first author.
"Predictive inference with Fleming–Viot-driven dependent Dirichlet processes." Bayesian Anal. 16 (2) 371 - 395, June 2021. https://doi.org/10.1214/20-BA1206