Semiparametric Bayesian Estimation in Latent Variable Models

我的博士学位目前的工作涉及到隐马尔可夫模型（hmm）的新理论的发展。这项工作是我和我的导师朱迪斯·卢梭共同完成的。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，其关键因素是后验分布，它量化了我们对模型参数的信念，一旦观察到数据，它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，它与数据一起决定了后验分布。先验分布本身有助于这一系列方法的灵活性，它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，在这些非参数模型中，贝叶斯方法的先验选择是一项微妙的任务。众所周知，推理可能过度依赖于这种选择，随着模型的增长，这种选择变得越来越难以解释——因此，将先验作为“专家先验意见”的经典解释越来越不有效。这标志着这一领域对理论结果的明确需求，这将在参数空间难以理解时为先验的选择提供信息，以确保将所得后验分布作为推理工具的应用建立在坚实的数学基础上。我们正在开发理论，为这些模型中的估计过程提供保证——具体来说，我们目前正在为hmm的“伯恩斯坦-冯-米塞斯”结果而努力，这将为预先选择提供信息，为上述两个目标的后验使用提供理论上的合理性。在这一理论的发展中，我们还将展示该模型的估计程序的存在性，这些估计程序本身也将被证明在理论上是有效的。我们将研究的结果被称为“半参数”，因为它们涉及在较大（非参数）模型中对较小参数的估计，并且是hmm上下文中的第一个此类结果。在未来，我们将开始与J. Rousseau的另一名学生合作，发展霍克斯过程的新理论。这些“自激”过程有着广泛的应用，从地震学、金融学到神经科学，甚至流行病学。只是在最近几年，它们的理论性质才开始被理解，但仍然有许多有趣和重要的理论问题，人们可以提出与这些模型有关的问题。与我们对hmm的研究一样，本研究的目标将是发展伯恩斯坦-冯-米塞斯结果，为使用后验分布作为估计和不确定性量化工具提供理论保证。这项研究属于EPSRC数学科学主题，属于统计和应用概率范畴。

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.