Mathematical Sciences: Using Inference from Simulation to Improve Efficiency of Simulations

Gibbs采样器和Metropolis算法等迭代模拟方法近年来已成为统计分析中常用的工具，特别是在贝叶斯推理中产生的后验分布的计算中。我们考虑了利用迭代模拟并行运行的输出来推断目标分布和转移概率的方法。这些推论可以用来自适应地改变模拟算法以加快收敛速度。开发和评估这些方法涉及四个研究目标。首先，需要对马尔可夫链模拟的数学理论中的猜想进行检验，特别是关于目前为正态分布建立的一些结果的一般性。其次，必须开发特定的自适应算法，注意过渡概率的自适应改变不会违反收敛条件。第三，基于并行仿真的迭代推理方法最有效；我们计划建立一个异步并行处理设置，其中独立运行的并行进程将结果发送到单个中央处理器，该处理器将持续执行关于目标分布的推断，并定期向外发送命令以更改转换规则。第四，这些方法旨在加快应用贝叶斯层次模型中后验分布的计算速度。随机漫步的计算机模拟在科学和统计学的许多领域变得越来越有用。我们正在研究的加快计算速度的方法是基于基本的数学和统计思想，灵感来自于各种应用的统计计算，包括预测总统选举，分析抽样调查中的无反应，寻找影响家庭高氡水平的因素，以及模拟有毒化学物质在体内的流动。所有这些问题都已经被随机漫步计算机模拟解决了；计算速度的提高将使我们和其他人能够适应更复杂、更精确的数学模型。

9404305 Gelman Iterative simulation methods such as the Gibbs sampler and Metropolis' algorithm have recently become popular tools in statistical analysis, especially in the calculation of posterior distributions arising in Bayesian inference. We consider methods for using the output from parallel runs of iterative simulation to perform inference about the target distribution and the transition probabilities. These inferences can then be used to adaptively alter the simulation algorithm to speed convergence. Developing and evaluating these methods involves four research objectives. First, conjectures in the mathematical theory of Markov chain simulation--notably on the generality of some results so far established for the normal distribution--need to be tested. Second, specific adaptive algorithms must be developed, with care taken that adaptive altering of the transition probabilities does not violate the conditions for convergence. Third, the methods of inference from iterative simulations work most effectively when based on parallel simulations; we plan to set up an asynchronous parallel processing setup, in which the independently running parallel processes would send results to a single central processor that would be continually performing inference about the target distribution and periodically send commands outward to alter the transition rules. Fourth, the methods are intended to be applied to speed the computation of posterior distributions in applied Bayesian hierarchical models. Computer simulations of random walks have become increasingly useful in many areas of science and statistics. The methods we are studying to speed the computations, based on basic mathematical and statistical ideas, were inspired by statistical computation for a variety of applications, including forecasting Presidential elections, analyzing nonresponse in sample surveys, searching for the factors that influence high radon levels in homes, and modeling the flow of toxic chemicals in the body. All of these problems have been attacked with random walk computer simulations; improvements in computation speed will allow us and others to fit more complicated and accurate mathematical models.