Drawing Exact Samples: Rejection Sampling, Density Fusion and Constrained Disaggregation

Sampling is an important topic in the area of computational statistics. Being able to draw samples from a designated distribution allows one to numerically compute various statistics without the need to solve for solutions analytically. A popular branch of the sampling method generates samples by evolving a stationary Markov chain that admits the target distribution as its stationary distribution. The problem, however, is that one does not have a universal criterion to assess whether the chain is stationary. On the other hand, exact simulation methods, being the focus of this thesis, always produce samples that precisely follow the target distribution. We first begin with the path-space rejection sampling for the exact simulation of diffusion bridges and show how this rejection scheme can be further set up into an exact simulation method for sampling product densities. We provide guidance on how to tune the algorithm parameters in order to attain a near-optimal performance and introduce the construction of an importance sampler/particle filter based on the same theoretical result for better efficiency. Finally, we show a variant of the sampler that deals with linear constraints which render most of the target distributions intractable. Two application studies are conducted in the end to demonstrate the effectiveness of the algorithm.