Bayesian Methods, Computation, Model Selection and Goodness of Fit with Complex Data

我的主要研究兴趣在于广泛的贝叶斯方法和计算复杂的模型，集成了建模和计算方面。在我的研究计划中，我开发，研究和实施方法，为来自健康，环境，体育和社会科学的大型或复杂数据提供统计推断。在这项研究计划中，我特别感兴趣的是为来自社交网络、环境、生态和健康系统的复杂数据类型开发新的贝叶斯方法。我们正处于数据科学时代，每个学科都有大量的数据，具有各种复杂性。这些复杂性可能是不完整性、维度、复杂结构或大数据的形式。在为这些数据开发统计模型时，我们通常包含一组关于数据生成的统计假设，无论是不确定的未来数据还是已经观察到的数据。这涉及到以结构化的方式将一组参数与数据联系起来。这可以使用参数或非参数模型来实现。在参数模型中，参数在有限维空间中，而在非参数模型中，参数在无限维空间中。请注意，在这两种方法中，参数都假定为固定未知数。在贝叶斯模型中，我们假设这些参数来自它们自己的分布，称为先验分布。先验分布用于量化我们关于参数的先验知识。当数据可用时，我们可以在给定数据的情况下使用参数的条件分布更新我们的先验知识。 ** 我将开发生成贝叶斯分层模型，其中参数空间位于有限维中。然后，我将考虑非参数贝叶斯模型，其参数空间具有无穷维。为了定义非参数贝叶斯模型，我们必须定义无限维空间上的概率分布。这些模型可用于将分布建模为简单分布的混合物，并识别可以解释变量之间复杂依赖关系的潜在类。这允许使用可计数的无限数量的混合物，这绕过了确定有限混合物模型中的组分的正确数量的需要。当模型基于共轭先验分布时，通过吉布斯采样从组分分布和/或混合组分的关联的参数的后验分布进行采样是可行的。当它们不共轭时，我将开发马尔可夫链蒙特卡罗（MCMC）抽样方法来了解模型参数。还将制定模型选择和拟合优度方法。该方法对预测哈德逊湾和加拿大北极东部地区的在线社会网络结构、新生儿安全和儿童健康、环境行为等具有一定的参考价值。********

My primary research interests lie broadly in Bayesian methods and computation for complex models which integrate both modelling and computational aspects. In my research program, I develop, study, and implement methods for providing statistical inference for large or complicated data arising from health, environmental, sports and social sciences. In this research proposal, I'm particularly interested in developing novel Bayesian methods for complex data types arising from social networks, environmental, ecological and health systems.******We are in the era of data science where enormous amounts of data arise in every discipline with various complexities. These complexities could be in the form of incompleteness, dimensionality, complex structures or big data. In developing a statistical model for these data, we typically embody a set of statistical assumptions concerning the generation of data, either uncertain future data or already observed data. This involves linking a set of parameters to the data in a structural fashion. This can be achieved using a parametric or nonparametric model. In a parametric model, the parameters are in finite-dimensional spaces while the parameters are in infinite-dimensional spaces in a nonparametric model. Note that the parameters are assumed to be fixed unknowns in both approaches. In Bayesian models, we assume that these parameters arise from their own distributions called prior distributions. A prior distribution is used to quantify our prior knowledge about the parameters. When data are available, we can update our prior knowledge using the conditional distribution of parameters, given the data. ******I will develop generative Bayesian hierarchical models where parameter spaces lie in finite dimensions. I will then consider nonparametric Bayesian models whose parameter space has infinite dimension. To define a nonparametric Bayesian model, we have to define a probability distribution on an infinite dimensional space. These models are useful to model distributions as mixtures of simpler distributions and to identify latent classes that can explain the complex dependencies between variables. This allows using a countably infinite number of mixtures, which bypasses the need to determine the correct number of components in a finite mixture model. When models are based on conjugate prior distributions, sampling from the posterior distribution of the parameters of the component distributions and/or of the associations of mixture components is feasible through Gibbs sampling. When they are not conjugate, I will develop Markov chain Monte Carlo (MCMC) sampling methods to learn about the model parameters. Model selection and goodness-of-fit methods will also be developed. The methods developed here will be useful to predict the online social network structures, newborn safety and child health, environmental behavior in Hudson Bay and Eastern Canadian Arctic. ********