Graphical models: discrete, Gaussian, coloured, estimation and model selection

我们建议的研究集中在开发统计推理工具，特别是用于选择图形模型。图形模型是统计学家工具箱中必不可少的一部分。它们在各种领域都有应用：在医学上用来模拟某些基因突变与疾病存在之间的相关性，或在金融学上用来模拟不同股票之间的依赖关系。*图形模型是通过图形来模拟不同变量之间的依赖关系的多元统计模型。图的顶点代表变量，而顶点之间的边是这些变量之间的依赖或独立的代码。考虑的变量可以是连续的(高斯)或离散的(多项式)，而被认为代表依赖关系的图可以是有向的或无向的。在我们的研究中，我们将考虑四种类型的图模型：高斯有色无向图、离散有向无环图(简称DAG)、离散无向图和离散异质图。每种类型的图最适合表示特定的数据集。我们的研究旨在选择一个最能代表给定数据集的模型，用于解释和/或预测。*1.有色高斯无向模型。这些都是经典的图形模型，在给定的变量组之间的关系上增加了等式限制。附加的约束减少了我们模型中的自由参数的数量。虽然这是一件好事，因为我们需要估计的参数更少，但这使得经典的图形高斯方法无法应用。我们的主要目的是给出一种新的基于生灭过程的贝叶斯模型选择方法。一个出了名的困难任务是在这样的模型空间中进行模型选择。这项任务很困难，因为许多图可以表示变量之间的相同依赖关系(我们说这样的模型是马尔可夫等价的)。马尔可夫等价DAG的集合可以由称为本质图的图来表示。利用我们最近的一些结果，我们希望将搜索归结为在本质图空间中的搜索。*3.MTP2离散对数线性模型。这些都是对数线性模型，变量之间存在正相关性。我们建议通过对模型参数的极大似然估计来进行模型选择。*4.离散异质模型被用来模拟k个子总体中给定变量之间的依赖关系。我们想要找出这些子群体背后的k个图之间的异同。高维离散异质模型还没有从贝叶斯的角度进行研究。我们截然不同的方法建议使用k个概率表的Parafac因式分解。

Our proposed research concentrates on developing tools of statistical inference, particularly for the selection of graphical models. Graphical models form an essential part of the statistician's toolbox. They find applications in various areas: in medicine to model the dependence between certain gene mutations and the existence of a disease, or in finance to model the dependence relationship between various stocks.******Graphical models are multivariate statistical models that model the dependence relationship between different variables by means of a graph. The vertices of the graph represent the variables while the edges between the vertices are a code for the dependence or independence between these variables. The variables considered can be continuous (Gaussian) or discrete (multinomial) and the graphs considered to represent the dependence relationship can be directed or undirected. In our research, we will consider four types of graphical models: Gaussian coloured undirected graphs, discrete directed acyclic graphs (abbreviated DAG), discrete undirected graphs and discrete heterogeneous graphs. Each type of graph is best adapted to represent certain data sets. Our research is aimed at selecting a model best representing a given data set, for the purpose of explanation and/or prediction.******1. Coloured Gaussian undirected models. These are classical graphical models with added equality restrictions on the relationship between given groups of variables. The additional constraints diminish the number of free parameters in our model. While this is a good thing because we have less parameters to estimate, it renders the classical graphical Gaussian methods impossible to apply. Our main aim is to give a new method of Bayesian model selection based on a process called a birth and death process.*********2. DAG discrete models. A notoriously hard task is to do model selection in the space of such models. This task is hard because many graphs can represent the same dependence relationship between variables (we say that such models are Markov equivalent). The set of Markov equivalent DAGs can be represented by a graph called an essential graph. Using some recent result of ours, we want to reduce the search to a search in the space of essential graphs.******3. MTP2 discrete loglinear models . These are log-linear models with positive associations between the variables. We propose to do model selection through maximum likelihood estimation of the parameter of the model.******4. Discrete heterogeneous models are used to model the dependence relationship between given variables for k different subpopulations. We want to identify the similarities and differences between the k graphs underlying the subpopulations. High-dimensional discrete heterogeneous models have not been studied from a Bayesian perspective. Our radically different approach proposes to use the Parafac factorization of the k tables of probabilities.******************