Bayesian Analysis of Elapsed Times in Continuous-Time Markov Chains

We explore Bayesian analysis for continuous-time Markov chain (CTMC) models based on a conditional reference prior. For CTMC models, inference of the elapsed time between chain observations depends heavily on the rate of decay of the prior as the elapsed time increases. Moreover, improper priors on the elapsed time may lead to improper posterior distributions. In addition to the elapsed time, an infinitesimal rate matrix also characterizes the CTMC. Usually, experts have good prior knowledge about the parameters of the infinitesimal rate matrix, and thus can provide well-informed priors. We show that the use of a proper prior for the rate matrix parameters together with the conditional reference prior for the elapsed time yields a proper posterior distribution. Finally, we demonstrate that, when compared to analyses based on priors previously proposed in the literature, Bayesian analysis on the elapsed time based on the conditional reference prior possesses better frequentist properties. The conditional reference prior therefore represents a better default prior choice for widely-used estimation software.