Mathematical Sciences: Topics in Discrete Probability and Algorithms

点击翻译按钮获取中文摘要

The aim of the project is to explore mathematically interesting probability questions arising directly or indirectly from the study of algorithms and combinatorics. One aspect concerns weak convergence methods for studying size-asymptotics of random combinatorial objects. Another aspect is a study of random walk in random environment on trees as a simple case of the Metropolis algorithm. A third part concerns algorithmic aspects of Bayesian posteriors on probability distributions given i.i.d. data, where we take a nonparametric prior derived from superprocess excursion. Random trees occur in various guises throughout the proposal. One part of the project is a theoretical study of how to approximate large random discrete structures by continuous structures, in contexts where such an approximation is rather subtle. Another part is a theoretical study of the most widely-used algorithm for computer simulation of complicated random objects, in a simplified setting. A final part is the development of sophisticated algorithms for updating statistical estimates of spatial distributions as more data becomes available.