On a Dirichlet Process Mixture Representation of Phase-Type Distributions

Daniel Ayala,Luis Gutiérrez,Leonardo Jofré,Ramsés H Mena

doi:10.1214/21-ba1272

Abstract

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.

Highlights

The Dirichlet process mixture (DPM) model can be defined as the random density model fP (y) = K(y | ξ) P, (2)
We present an explicit representation of phase-type distributions as an infinite mixture of Erlang distributions
We demonstrated a clear connection between phase-type distributions, mixtures of Erlang distributions, and Bayesian nonparametric inference in this work

Summary

Introduction

Πp), with πi = P(X0 = i), be a row vector in the p-dimensional simplex space Sp. A phase-type distribution with dimension p is defined as the time until absorption of the Markov jump process {Xt}t≥0, with initial distribution π and subintensity matrix T, namely the distribution of the random variable Y := inf {t > 0 | Xt = p + 1}. Inference for Bayesian nonparametric mixture models, is nowadays relatively standard (see, e.g., Escobar and West, 1995; Neal, 2000; Ishwaran and James, 2001; Walker, 2007; Kalli et al, 2011; Miller and Harrison, 2018), whereas inference for phase-type distributions is still challenging task (Bladt et al, 2003; Aslett, 2012) Both classical and Bayesian approaches to phase-type distribution inference available in the literature, resort to the underlying Markov jump processes.

A SPH-distribution representation via Erlang kernels

Posterior inference

Phase-type representation

Monte Carlo study

Comparison with other inference approaches

Renewal function estimation

Application to real data sets

Discussion and concluding remarks