A host of sequential models in probability and statistics are characterized by time reversibility, from Markov chain Monte Carlo samplers to queueing networks. In physics, this property arises naturally from Hamiltonian mechanics. Molecular dynamics simulations are computer experiments which approximate classical mechanics in a system of interacting particles; in consequence, they are frequently reversible. Recent technical progress has made it possible to investigate the dynamics of biological macromolecules in silico using molecular dynamics simulations. An active area of research within this field is concerned with modeling the output of a simulation stochastically. This dissertation deals with the problem of incorporating knowledge of reversibility into the estimation and testing of stochastic models. We define a range of Bayesian inference algorithms, which are motivated by specific problems in the analysis of molecular dynamics simulations.