## Abstract

Phase-type (PHT) distributions provide a natural model for a wide range of stochastic processes where the event of interest is first passage time to a given state or set of states. However, they present interesting inferential challenges. Whilst both general likelihood and Bayesian approaches have been developed in the context of distribution fitting, there has been relatively little work on inference where there is a scientific interpretation of the underlying stochastic process. It is this latter situation which we address in the current work.

Our work builds on the Markov chain Monte Carlo (MCMC) algorithm developed by Bladt *et al.* (2003) in a number of directions. Firstly, in order to facilitate models in which the stochastic process has some scientifically interesting interpretation the conjugacy properties are shown to hold when constraints are imposed on the structure of the underlying Markov process, both in terms of prohibiting certain state transitions and of fixing certain rate parameters as being equal. The existing algorithm is also shown to naturally incorporate right-censored observations as can be common in real world temporal data.

These modifications can lead to computational issues in some situations and alleviating these in a wide class of situations is the second contribution of this work. The original algorithm is adapted to improve the step involving simulation of the Markov jump process underlying the PHT, which is required for each observation.

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## Citation Information

Please cite this proceedings paper as:

Aslett, L. J. M. and Wilson, S. P. (2011), Markov chain Monte Carlo for Inference on Phase-type Models, *in* ‘ISI 2011 Proceedings’.

BibTeX:

```
@inproceedings{Aslett2011,
title={Markov chain Monte Carlo for Inference on Phase-type Models},
author={Aslett, L. J. M. and Wilson, S. P.},
year={2011},
booktitle={ISI 2011 Proceedings}
}
```

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