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Vol.:(0123456789)

Computational Statistics

https://doi.org/10.1007/s00180-022-01273-w

1 3

ORIGINAL PAPER

On theestimation ofpartially observed continuous‑time

Markov chains

AlanRiva‑Palacio1 · RamsésH.Mena1· StephenG.Walker2

Received: 17 December 2020 / Accepted: 5 August 2022

© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022

Abstract

Motivated by the increasing use of discrete-state Markov processes across applied

disciplines, a Metropolis–Hastings sampling algorithm is proposed for a partially

observed process. Current approaches, both classical and Bayesian, have relied on

imputing the missing parts of the process and working with a complete likelihood.

However, from a Bayesian perspective, the use of latent variables is not necessary

and exploiting the observed likelihood function, combined with a suitable Markov

chain Monte Carlo method, results in an accurate and eﬃcient approach. A com-

prehensive comparison with simulated and real data sets demonstrate our approach

when compared with alternatives available in the literature.

Keywords Bayesian estimation· Transition matrix· Credit risk scoring

1 Introduction

We consider the inference problem of a partially observed continuous-time Markov

chain (CTMC), written as

X∶= {

X

(

t

);

t≤𝜏

}

, that take values on a ﬁnite state space,

𝕊∶= {1, …,m}

. Such continuous-time discrete-state systems ﬁnd applications in

areas such as physics, Van Kampen (2007); ecology, Fukaya and Royle (2013);

neuroscience, Sauer (2016); and ﬁnance, Pardoux (2008). Hence, the need for

The authors gratefully acknowledge the support of project CONTEX 2018-9B and PAPIIT-UNAM

IG100221.

* Alan Riva-Palacio

alan@sigma.iimas.unam.mx

Ramsés H. Mena

ramses@sigma.iimas.unam.mx

Stephen G. Walker

s.g.walker@math.utexas.edu

1 IIMAS, UNAM, MexicoCity, Mexico

2 University ofTexas atAustin, Austin, USA

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