【作者】 张晨；

【导师】 吴茜茜；

【作者基本信息】 合肥工业大学 ， 概率论与数理统计， 2019， 硕士

【摘要】 贝叶斯统计推断方法作为统计学研究中的重要分支之一,它在各个研究领域有着广泛的应用。在贝叶斯统计推断过程中,如何选择并计算似然函数是最核心问题。特别地,在生物数学领域中,随着统计模型的不断发展,模型的似然函数越来越复杂,往往不具有解析表达式或者过于复杂导致无法计算,基于复杂似然函数的贝叶斯推断很困难。此时,近似贝叶斯计算(Approximate Bayesian Computation,ABC)方法可以很好地解决这一问题。本文将从理论分析,算法设计以及实际应用这三个方面对目前常见的近似贝叶斯计算方法进行了深入的探讨。本文针对各个算法进行了详实的比较,提出了一种基于粒子滤波器和蒙特卡洛方法的模型选择算法,并应用于系列SIR模型选择问题。本文涵盖的具体研究内容如下:1.近似贝叶斯计算的基本原理的描述以及讨论实现过程中汇总统计量的选取和容差阈值的界定;2.传统的ABC拒绝算法、ABC回归算法,以及基于马尔科夫链蒙特卡洛(Markov Chain Monte Carlo,MCMC)方法的ABC-MCMC算法和基于粒子滤波器的ABC部分拒绝控制(partial rejection control,PRC)算法、ABC种群蒙特卡洛(population Monte Carlo,PMC)算法、ABC序贯蒙特卡洛(sequential Monte Carlo,SMC)算法的介绍和比较;3.基于粒子滤波器和蒙特卡洛方法,提出了一种ABC-PMC模型选择算法;4.参数估计和模型选择的应用实例:二项分布模型,泊松分布模型,流感病毒传染模型以及经典传染病模型选择问题。更多还原

【Abstract】 Being one of the most important branches of statistical studies,Bayesian statistical inference method has been applied in various research fields.In the process of Bayesian statistical inference,how to select and calculate the likelihood function is the key question.Particularly,with the continuous development of statistical models in the field of biological mathematics,the likelihood functions of models become more and more complex and it is often hard to obtain analytical expressions or to be calculated.Therefore,it is difficult to deal with Bayesian inference due to the complexity of likelihood functions.At this stage,approximate Bayesian computation(ABC)can be applied to solve the problem.In this work,we will discuss the approximate Bayesian computation thoroughly from three aspects,namely,the theoretic analyses,algorithm designs and practical applications.Based upon the detailed comparisons between various algorithms,we have proposed a novel algorithm that helps with model selections based on the particle filtering and Monte Carlo method.And it has been applied to the model selection problem for SIR models,which reveals some good results.The overall contents of this work are as follows:1.Descriptions of the basic principle of approximate Bayesian computation and the selection of the summary statistics and the definition of tolerance threshold during the realization;2.Descriptions of the basic ABC rejection algorithm,ABC regression algorithm,ABC-MCMC algorithm based on MCMC method,ABC-PRC algorithm based on particle filtering,ABC-PMC algorithm,ABC-SMC algorithm and comparisons between these algorithms;3.Descriptions of the proposed ABC-PMC model selection algorithm based on particle filter and Monte Carlo method;4.Applied examples in parameter estimation and model selection: a binomial distributed model,a Poisson distributed model,an influenza virus infection model and a classical infectious disease model.更多还原