Abstract
Rao and Teh (2012, 2013) introduced an efficient MCMC algorithm for sampling from the posterior distribution of a hidden Markov jump process. The algorithm is based on the idea of sampling virtual jumps. In the present paper we show that the Markov chain generated by Rao and Teh’s algorithm is geometrically ergodic. To this end we establish a geometric drift condition towards a small set. A similar result is also proved for a special version of the algorithm, used for probabilistic inference in Continuous Time Bayesian Networks.
Highlights
Markov jump processes (MJP) are natural extensions of Markov chains to continuous time
Such a graphical representation allows for decomposing a large intensity matrix into smaller conditional intensity matrices
Miasojedow and Niemiro (2016) proved geometric ergodicity of Rao and Teh’s algorithm in a special case of the homogeneous MJPs observed at discrete moments and when the virtual jumps are introduced by uniformization procedure
Summary
Markov jump processes (MJP) are natural extensions of Markov chains to continuous time. To the best of our knowledge the most general efficient method for a finite state space is that proposed by Rao and Teh (2013), and extended to a more general class of continuous time discrete systems in Rao and Teh (2012) Their algorithm is based on introducing so-called virtual jumps and a thinning procedure for Poisson processes. Miasojedow and Niemiro (2016) proved geometric ergodicity of Rao and Teh’s algorithm in a special case of the homogeneous MJPs observed at discrete moments and when the virtual jumps are introduced by uniformization procedure. Note that in practice the parameters of the hidden MJP may be unknown and have to be estimated For both Bayesian and frequentist statistical inference Rao and Teh’s algorithm can be applied as a part of more complex algorithms. Set {n, n + 1, . . . , m} is denoted by [n : m] (for integer n ≤ m)
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