Mathematical Sciences: Topics in Discrete Probability and Algorithms

该项目的目的是探索直接或间接产生于算法和组合学研究的数学上有趣的概率问题。一方面涉及研究随机组合对象尺寸渐近性的弱收敛方法。另一个方面是研究树上随机环境中的随机游动，作为Metropolis算法的一个简单例子。第三部分涉及关于给定I.I.D.的概率分布的贝叶斯后验的算法方面。数据，其中我们取来自超过程漂移的非参数先验。在整个提案中，随机树以各种形式出现。该项目的一个部分是关于如何用连续结构来逼近大型随机离散结构的理论研究，在这种逼近相当微妙的情况下。另一个部分是在简化的设置下，对复杂随机对象的计算机模拟最广泛使用的算法的理论研究。最后一部分是开发复杂的算法，随着更多数据的出现，更新空间分布的统计估计。

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.