Randomized Quasi-random Importance Resampling Method

With the development of computer technology,Monte Carlo methods are widely used in financial engineering,biomedicine,macroeconomics,statistical physics and other fields.Sampling from a target distribution is crucial for Monte Carlo methods,so it is a challenge to finding efficient procedures to derive samples.The sampling/importance resampling(SIR)method and the Gibbs sampler are two representative strategies.The former obtains independent samples by sampling from a proposal that is different from the target distribution,while the latter produces a Markov Chain whose stationary distribution is the target distribution.In recent decades,considerable modifications have been made to improve the efficiency of sampling or to broaden its range of applications.The SIR method is sensitive to proposal distribution,but it is difficult to derive the optimal or even suboptimal proposal in the sense of variance.The Quasi-random sampling/importance resampling(QSIR)method improves sample quality by engaging SIR with Quasi-Monte Carlo.However,the QSIR is bound,because it is not applicable to proposals with a dependent structure.And samples derived by QSIR are deterministic,which can not be used to estimate the error.Otherwise,SIR and Quasi-Monte Carlo methods are more suitable for low-dimensional distributions.This paper concentrates on the sampling/importance resampling algorithm.First we introduce the idea of randomization in Quasi-Monte Carlo into the sampling/importance resampling algorithm and put forward the randomized Quasi-random sampling/importance resampling algorithm(RQSIR).Furthermore,the validity of this new algorithm is proved,and the convergence speed is provided.The Rosenblatt transformation,which could be operated on proposals with dependent structure,is applied to get a pool of alternative samples.Then,it makes the QSIR could be applied more widely,and samples obtained by RQSIR are employed to estimate the error.Second,the global sampling/importance resampling(GSIR)is proposed for multi-modal distributions.we give an algorithm called"grid grower" to get the domain which covers the support of target distribution as much as possible,and then construct a randomized Quasi-Monte Carlo point set in this area.Besides,simulation results show that this layout performs better for drawing samples from multi-modal distribution than other classical techniques,for instance,the Evolutionary Monte Carlo method and global likelihood ratio sampler.Finally,due to the inadequacy of Quasi-Monte Carlo techniques for high-dimensional problems,a hybrid Gibbs sampler is given by incorporating RQSIR or GSIR with Gibbs sampler.From the simulation results,it can be seen that the hybrid Gibbs sampler applies to high-dimensional case and non-conjugate Bayesian problems.A brief introduction to each chapter of this dissertation is given as follows.Chapter one is intended to be an introduction.It briefly introduces the background of this topic,a comprehensive review and problems which will be explored in this dissertation are also presented here.Finally,it describes the structure and innovations of this article.Chapter two shows some necessary preliminary materials.It presents the details of Quasi-Monte Carlo and sampling/importance resampling method,which help go through the work of the following chapters.Chapter three proposes the randomized Quasi-random sampling/importance resampling algorithm,and also proves the convergence of the proposals to the target while the accuracy of estimation is also testified.Based on Quasi-random sampling/importance resampling algorithm,we bring the idea of randomization Quasi-Monte Carlo into it,and make a randomization version.Chapter four affords a easy way to produce the pool of alternative samples.This way saves the cost of calculation and also makes the new algorithm more applicable.The simulation results show that this new procedure performs well for sampling from multi-modal density.Chapter five presents a hybrid Gibbs procedure.This procedure focus on sampling from high-dimensional distribution,which essentially integrates the RQSIR within Gibbs sampler just like other within Gibbs sampler.Also,this hybrid Gibbs sampler could be used for non-conjugate bayesian model.Numerical results show that this hybrid Gibbs sampler performs well in both high-dimensional cases and non-conjugate Bayesian problems.Chapter six summarizes the work of this dissertation and looks forward to future study.