| Bridge sampling is among the most effective Monte Carlo methods for estimating ratios of normalizing constants of probability densities. It requires only (1) that it be possible to make random draws from each of the probability distributions, and (2) a bridge function. There are few requirements that the bridge function must satisfy, but a bad choice can have injurious effects on the estimator's efficiency. The optimal bridge function is known for the case when the random draws are statistically independent. In many problems, researchers must rely on sampling methods that produce dependent draws, such as those from Markov chains. The optimal bridge function for the dependence case is generally unknown and is considered an open problem. This dissertation introduces methods specifically designed to treat the dependent case, and applies them to the problem in statistical physics of estimating free energy differences of the Ising model.; Among the contributions of this thesis are: proofs (often taken for granted) of some of the basic claims of bridge sampling, a procedure for choosing a sensible bridge function in the dependence case, methods for redefining draws to reduce errors, a statistical primer of the Ising model, a result relating the physical notion of criticality to the probabilistic notion of Hellinger distance, and a review of techniques for simulating the Ising model, including the implementation of an exact bounding chain sampler. Additionally, we show how this exact sampler can be used to determine critical points of physical systems. |
