An explicit representation of phase-type distributions as an infinite mixture of Erlang distributions is introduced. The representation unveils a novel and useful connection between a class of Bayesian nonparametric mixture models and phase-type distributions. In particular, this sheds some light on two hot topics, estimation techniques for phase-type distributions, and the availability of closed-form expressions for some functionals related to Dirichlet process mixture models. The power of this connection is illustrated via a posterior inference algorithm to estimate phase-type distributions, avoiding some difficulties with the simulation of latent Markov jump processes, commonly encountered in phase-type Bayesian inference. On the other hand, closed-form expressions for functionals of Dirichlet process mixture models are illustrated with density and renewal function estimation, related to the optimal salmon weight distribution of an aquaculture study.
The work of the first author was supported by “Becas Doctorado Nacional CONICYT 2017 Folio No. 21171601”. The work of the third author was supported by “Proyecto REDES ETAPA INICIAL Convocatoria 2017 REDI170094” and by ANID–Millennium Science Initiative Program–NCN17_059. The fourth author acknowledges the support of CONTEX project 2018-9B.
The authors are grateful for the valuable comments made by two anonymous referees, the Associate Editor and the Editor. The authors thank professor Ricardo Olea Ortega by providing the salmon weights data set.
"On a Dirichlet Process Mixture Representation of Phase-Type Distributions." Bayesian Anal. Advance Publication 1 - 26, 2021. https://doi.org/10.1214/21-BA1272