Semiparametric Bayesian Estimation in Latent Variable Models

目前的工作，我的哲学博士涉及的新理论的发展隐马尔可夫模型（HALTH）。这项工作是我和我的上司朱迪思·卢梭合作完成的。这些模型的应用范围很广，包括信号处理、自然语言处理、经济学、分子动力学和生物化学。在现代统计应用中，贝叶斯非参数方法因其灵活性而无处不在，其关键元素是后验分布，一旦数据被观察到，它就量化了我们对模型参数的信念，并且其允许点估计和不确定性量化。这种推理方法的基本要素之一是先验分布，它与数据一起构成了这种后验分布。先验分布本身有助于这一系列方法的灵活性，通过允许我们以分层的方式构建模型，并允许我们通过惩罚过于复杂的参数来更直接地解释数据生成过程，然而，在贝叶斯方法中先验的选择在这些非参数模型中是一项微妙的任务。人们已经知道，推理可能过度依赖于这种选择，随着模型的发展，这种选择变得越来越不可解释--因此，将先验解释为“专家先验意见”的经典解释越来越不有效。这标志着一个明确的需要在这一领域的理论结果，这将通知先验的选择时，参数空间是难以理解的，以确保所产生的后验分布作为一个推理工具的应用程序是放在一个坚实的数学基础。我们正在开发的理论，这将提供保证，估计程序在这样的模型-具体来说，我们目前正致力于为Hyndrome提供一个“伯恩斯坦冯-米塞斯”的结果，它将为先前的选择提供信息，从而为上述两个目的的后验的使用提供理论上的可靠性。在这个理论的发展，我们也将显示存在的估计程序，这个模型，这本身也将被证明是理论上有效的。我们将努力的结果被称为“semiparametric”，因为它们涉及到一个更大的（非参数）模型内的一个较小的参数估计，并且是第一个在Hessel context.In未来，我们将开始工作与J.卢梭的另一个学生合作，发展的新理论霍克斯过程。这些“自激”过程有着广泛的应用，从地震学到金融学，到神经科学，甚至到流行病学，只是在最近几年，人们才开始了解它们的理论特性，但仍然存在着许多与这些模型有关的有趣而重要的理论问题。与我们的工作Hessel，本研究的目标将是发展伯恩斯坦冯-米塞斯结果提供理论保证的后验分布作为一种工具，估计和不确定性quantified.This研究属于福尔斯下的EPSRC数学科学的主题，在统计和应用概率的类别。

The current work for my DPhil involves the development of novel theory for Hidden Markov Models (HMMs). This work is undertaken collaboratively between my supervisor, Judith Rousseau, and myself. The applications of such models are wide-ranging, including signal processing, natural language processing, economics, molecular dynamics and biochemistry.In modern statistical applications, Bayesian nonparametric approaches are ubiquitous for their flexibility, their key element being the posterior distribution which quantifies our beliefs about the parameters of the model once the data has been observed, and which allows for point estimation and uncertainty quantification. One of the fundamental elements of this method of inference is the prior distribution, which alongside the data informs this posterior distribution. The prior distribution itself contributes to the flexibility of this family of methods, by allowing us to construct models in a hierarchical way, and by allowing us to favour more straightforward explanations of the data generating process by penalising overly complicated parameters.However, the choice of prior in Bayesian methods is a delicate task in these nonparametric models. It is already known that inference can be overly dependent on this choice, which becomes less and less interpretable as the model grows - thus making the classical interpretation of the prior as an `expert prior opinion' less and less valid. This marks a clear need for theoretical results in this area, which will inform the choice of priors when the parameter space is difficult to understand, to ensure that the application of the resulting posterior distribution as an inferential tool is placed upon a firm mathematical foundation.We are developing theory which will provide guarantees for estimation procedures in such models - specifically, we are currently working towards a `Bernstein von-Mises' result for HMMs which will inform prior choice to provide theoretical soundness to the use of the posterior for both of the aforementioned ends. In the development of this theory, we will also show the existence of estimation procedures for this model, which will themselves also be shown to be theoretically valid. The results which we will work towards are termed `semiparametric', as they involve the estimation of a smaller parameter within a larger (nonparametric) model, and are the first of their kind in the HMMs context.In the future, we will begin work in collaboration with another of J. Rousseau's students, on the development of novel theory for Hawkes processes. These `self-exciting' processes have wide-ranging applications from seismology, to finance, to neuroscience and even to epidemiology.It is only in recent years that their theoretical properties have begun to be understood, but there still remains a wide range of interesting and important theoretical questions which one can ask in relation to these models. As with our work for HMMs, the goal of this research will be to develop Bernstein von-Mises results which provide theoretical guarantees for the use of the posterior distribution as a tool for estimation and uncertainty quantification.This research falls under the EPSRC theme of Mathematical Sciences, in the category of Statistics and Applied Probability.