4.4 Article

On a Dirichlet Process Mixture Representation of Phase-Type Distributions

Journal

BAYESIAN ANALYSIS
Volume 17, Issue 3, Pages 765-790

Publisher

INT SOC BAYESIAN ANALYSIS
DOI: 10.1214/21-BA1272

Keywords

Bayesian nonparametrics; Erlang distribution; mixture model; renewal function

Funding

  1. Becas Doctorado Nacional CONICYT 2017 [21171601]
  2. Proyecto REDES ETAPA INICIAL Convocatoria 2017 [REDI170094]
  3. ANID-Millennium Science Initiative Program [NCN17 059]
  4. CONTEX project [2018-9B]

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This paper introduces an explicit representation of phase-type distributions as an infinite mixture of Erlang distributions, revealing a novel and useful connection between a class of Bayesian nonparametric mixture models and phase-type distributions. The paper explores estimation techniques for phase-type distributions and closed-form expressions for functionals related to Dirichlet process mixture models, and demonstrates the power of this connection through a posterior inference algorithm.
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 mod-els 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 al-gorithm 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.

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