TwoState Imprecise Markov Chains for Statistical Modelling of TwoState NonMarkovian Processes
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Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications, PMLR 103:394403, 2019.
Abstract
This paper proposes a method for fitting a twostate imprecise Markov chain to time series data from a twostate nonMarkovian process. Such nonMarkovian processes are common in practical applications. We focus on how to fit modelling parameters based on data from a process where time to transition is not exponentially distributed, thereby violating the Markov assumption. We do so by first fitting a manystate (i.e. having more than two states) Markov chain to the data, through its associated phasetype distribution. Then, we lump the process to a twostate imprecise Markov chain. In practical applications, a twostate imprecise Markov chain might be more convenient than a manystate Markov chain, as we have closed analytic expressions for typical quantities of interest (including the lower and upper expectation of any function of the state at any point in time). A numerical example demonstrates how the entire inference process (fitting and prediction) can be done using Markov chain Monte Carlo, for a given set of prior distributions on the parameters. In particular, we numerically identify the set of posterior densities and posterior lower and upper expectations on all model parameters and predictive quantities. We compare our inferences under a range of sample sizes and model assumptions.
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