Abstract
We consider the problem of estimating the transition rate matrix of a continuous-time Markov chain from a finite-duration realisation of this process. We approach this problem in an imprecise probabilistic framework, using a set of prior distributions on the unknown transition rate matrix. The resulting estimator is a set of transition rate matrices that, for reasons of conjugacy, is easy to find. To determine the hyperparameters for our set of priors, we reconsider the problem in discrete time, where we can use the well-known Imprecise Dirichlet Model. In particular, we show how the limit of the resulting discrete-time estimators is a continuous-time estimator. It corresponds to a specific choice of hyperparameters and has an exceptionally simple closed-form expression.
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Notes
- 1.
- 2.
The assumption \(d_x>0\) prevents division by zero in (3). However, \(n_{xy}\) might be zero and, if then also \(\alpha _{xy}=0\), the posterior cannot be normalised and will still be improper. Nevertheless, using an intuitive (but formally cumbersome) argument we can still identify this posterior for \(q_{xy}\) with the (discrete) distribution putting all mass at zero. Alternatively, we can motivate (3) by continuous extension from the cases where \(\alpha _{xy}>0\), similarly yielding the estimate \(\hat{q}_{xy}=0\) at \(\alpha _{xy}=n_{xy}=0\).
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Acknowledgements
The work in this paper was partially supported by H2020-MSCA-ITN-2016 UTOPIAE, grant agreement 722734. The authors wish to thank two anonymous reviewers for their helpful comments and suggestions.
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Krak, T., Erreygers, A., De Bock, J. (2019). An Imprecise Probabilistic Estimator for the Transition Rate Matrix of a Continuous-Time Markov Chain. In: Destercke, S., Denoeux, T., Gil, M., Grzegorzewski, P., Hryniewicz, O. (eds) Uncertainty Modelling in Data Science. SMPS 2018. Advances in Intelligent Systems and Computing, vol 832. Springer, Cham. https://doi.org/10.1007/978-3-319-97547-4_17
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