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

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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.