基于狄利克雷过程的潜变量模型贝叶斯半参数分析

Latent variable model (LVM) is an important statistical method for identifying the interrelationships among latent variables and building the relationships between latent constructs and observed variables. The statistical inferences based on the common parametric Bayesian approaches often suffer from serious biases when the underlying distrbutions deviate from the assumed forms or when data contain extrem values. In particular, a single parametric model is difficult to identify the dependence among the distributions at the different levels. In this study, we will develope Bayesian semiparametric analysis procedure for analyzing LVMs. Comparing to the existing developments of Bayesian semiparametric analysis for LVMs, the novelty of this study lies in the following four aspects: (i) we will intensively carry out semiparametric Bayesian analysis for multilevel, multigroup and nested LVMs based on the multi-Dirichlet processes; (ii) we will systematically develope Bayesian semiparametric analysis procedure for temporal-spatial LVMs; (iii) we will develope a Bayesian semiparametric analysis for LVMs with qualitative data; and (iv) we will focus on the Bayseian semiparametric model selection in terms of the likelihood of data. For Bayesian analysis, posterior inferences will be carried out through simulation-based methods such as Markov chains Monte Carlo algorithm and data-augmentation technique. Model fit will be assessed through model selections/comparison, statistical diagnosis and related hypothesis test procedures. More importantly, we will illustrate the effectiveness of the proposed methodology in the different application contexts.

潜变量模型是一种用来体现潜变量相互关系和构建潜变量与观测变量关联的重要统计方法。通常的参数贝叶斯推断在分布偏离假定形式或数据含有异常值时有严重偏差。特别地，对于多级、多组别、嵌套式潜变量模型，单一参数模型很难界定不同水平分布间的关联性。本研究着力于潜变量模型贝叶斯半参数分析。相较于目前该领域的研究进展，本研究创新之处为：（一）基于多狄利克雷过程对多级、多组别和嵌套潜变量模型集中展开半参数贝叶斯分析；（二）系统地展开时空潜变量模型的贝叶斯半参数分析；（三）对带有定性数据的潜变量模型进行贝叶斯半参数分析；（四）集中考虑基于数据似然的贝叶斯半参数模型选择。为了贝叶斯分析，我们将基于诸如马尔可夫链蒙特卡罗算法和数据扩充技术等仿真方法来进行后验推断。模型拟合将通过模型的选择/比较、统计诊断和相关的假设检验程序来达到。更重要地，我们将在不同的应用环境中展示方法的有效性。

潜变量模型是一种用来体现潜变量相互关系和构建潜变量与观测变量关联的重要统计方法。通常的参数贝叶斯推断在分布偏离假定形式或数据含有异常值时有严重偏差。特别地，对于多组别、嵌套式潜变量模型，单一参数模型很难界定不同水平分布间的关联性。我们着力研究了基于狄利克雷过程的潜变量模型贝叶斯半参数分析。内容包括:（1）我们对多水平潜变量模型的进行半参数贝叶斯分析，解决了模型参数在各个层面上的异质性;（2）我们系统地对隐马尔可夫动力潜变量模型的展开参数和半参数贝叶斯分析；(3）对带有次序性和连续型数据的潜变量模型进行贝叶斯半参数分析;（4）我们解决了有限维狄利克雷先验下数据边际似然的计算。（4）我们对有限维先验的有关性质作了讨论，得到了有限样本下“类”的特性。（5）我们首次解决了有限维狄利克雷先验下数据边际似然的计算。另外，我们还将当前潜变量模型延伸到了用来解决“零膨胀”的两部分潜变量模型并作了统计分析。在计算上，基于马尔可夫链蒙特卡罗算法和数据扩充技术等仿真方法对模型展开后验分析。解决了参数估计、模型拟合、模型选择/比较、统计诊断和相关的假设检验程序等诸多问题。更重要地，我们在不同的应用环境中展示研究的实用价值和方法的有效性。