Bayesian inference for Continuous-Time Markov Chains (CTMCs) on countably infinite spaces is notoriously difficult because evaluating the likelihood exactly is intractable. One way to address this challenge is to first build a non-negative and unbiased estimate of the likelihood -- involving the matrix exponential of finite truncations of the true rate matrix -- and then to use the estimates in a pseudo-marginal inference method. In this work, we show that we can dramatically increase the efficiency of this approach by avoiding the computation of exact matrix exponentials. In particular, we develop a general methodology for constructing an unbiased, non-negative estimate of the likelihood using doubly-monotone matrix exponential approximations. We further develop a novel approximation in this family -- the skeletoid -- as well as theory regarding its approximation error and how that relates to the variance of the estimates used in pseudo-marginal inference. Experimental results show that our approach yields more efficient posterior inference for a wide variety of CTMCs.

翻译：对连续时间标记链(CCTMCs)的可计算无限空间的贝叶斯推论非常困难,因为评估可能性的准确性是难以做到的。解决这一挑战的一个方法就是首先对可能性作出非消极和公正的估计 -- -- 涉及真实速率矩阵的有限变速指数矩阵,然后用假边推论法使用估计值。在这项工作中,我们表明,通过避免计算精确的矩阵指数,我们可大幅提高这一方法的效率。特别是,我们制定了一种通用方法,用以构建一种对使用双聚苯乙烯矩阵指数指数指数的可能性的不偏袒、非否定的估计。我们进一步在这个家庭里发展了一种新的近似性近似值 -- -- 类 -- 以及关于其近似误差的理论,以及这与伪边推论中所用估计值的差异有关的理论。实验结果表明,我们的方法为多种CTMC提供了更高效的远地点推断值。