Markov chain Monte Carlo for exact inference for diffusions

We develop exact Markov chain Monte Carlo methods for discretely-sampled, directly and indirectly observed diffusions. The qualification "exact" refers to the fact that the invariant and limiting distribution of the Markov chains is the posterior distribution of the parameters free of any discretisation error. The class of processes to which our methods directly apply are those which can be simulated using the most general to date exact simulation algorithm. The article introduces various methods to boost the performance of the basic scheme, including reparametrisations and auxiliary Poisson sampling. We contrast both theoretically and empirically how this new approach compares to irreducible high frequency imputation, which is the state-of-the-art alternative for the class of processes we consider, and we uncover intriguing connections. All methods discussed in the article are tested on typical examples.Comment: 23 pages, 6 Figures, 3 Table

Bridging trees for posterior inference on Ancestral Recombination Graphs

We present a new Markov chain Monte Carlo algorithm, implemented in software Arbores, for inferring the history of a sample of DNA sequences. Our principal innovation is a bridging procedure, previously applied only for simple stochastic processes, in which the local computations within a bridge can proceed independently of the rest of the DNA sequence, facilitating large-scale parallelisation.Comment: 23 pages, 9 figures, accepted for publication in Proceedings of the Royal Society