Random Measures: Asymptotics, Bayesian Inference, and Stochastic Dynamics

随机测量：渐进、贝叶斯推理和随机动力学

基本信息

批准号：
RGPIN-2016-05400
负责人：
Feng, Shui
金额：
$ 1.6万
依托单位：
McMaster University
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2019
资助国家：
加拿大
起止时间：
2019-01-01 至 2020-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=678851
关键词：
Random Measures Asymptotics Bayesian Inference

项目摘要

A random measure is a measure-valued random element. A collection of random measures becomes a measure-valued process. Random measures and measure-valued processes have been highly active research subjects over the past three decades in probability theory, stochastic processes, and statistics. Some of the well studied models include random partitions, coalescent, Dirichlet process, stick breaking model, Fleming-Viot process, and various urn models. One main motivation for their study comes from modelling complex interacting systems and the corresponding evolving dynamics. Intensive studies in this area have led to many theoretical progresses in probability theory and stochastic processes, and plenty of applications in astrophysics, chemistry, communication, ecology, economics, linguistics, machine learning, and population genetics. ***The progresses and advances obtained so far are nothing less than remarkable. But many challenges remain. Firstly, many random measures and processes depend on unknown parameters. Their asymptotic behaviours in various parametric regions are important features and mostly unknown. Secondly, in the study of stochastic dynamics, there is not only a strong demand in making the models closer to reality but also a clear need of calibration methods for data fitting. More efficient statistic methods need to be developed. Thirdly, mathematical models such as coalescent have been used in the study of the genealogical structure of an evolving population. But many existing models are exchangeable. An important challenge will be to develop mathematical models for non-exchangeable genealogical structures. Finally, random probability measures have been used as the building blocks of Bayesian inference. Models such as the Dirichlet process, stick breaking models, the Hierarchical Dirichlet process, Chinese restaurant process, and Indian buffet process have been developed and widely used in statistical inferences. Recent explosion in data accumulation exposes the limitations of these models in capturing real-world phenomena. More complex, manageable models are required. Mathematically one would need to develop models of random measures that involve strong local and long range interactions, and complicated spatial structures.***The proposed research will address these challenges through the development of new models of random measures and random processes, the design of efficient algorithms, and the analysis of asymptotic behaviour of various complex systems under different limiting regimes. The potential impact of this research is well beyond probability theory. Direct applications include, but not limited to, the species sampling issues in ecology, the genealogical structures in population genetics, spin glass models in physics, and portfolio theory in financial engineering. It will bring benefit to Canada both academically and economically. ********

随机度量是测量值的随机元素。一系列随机度量成为一个衡量值的过程。在过去的三十年中，在概率理论，随机过程和统计数据中，随机测量和评估值的过程一直是高度活跃的研究对象。一些经过良好研究的模型包括随机分区，融合，迪利奇过程，棍棒破坏模型，弗莱明·维奥特工艺和各种urn模型。他们研究的主要动机是对复杂的相互作用系统建模和相应的不断发展的动态。该领域的深入研究导致了概率理论和随机过程的许多理论进步，以及在天体物理学，化学，交流，生态，经济学，语言学，机器学习和人口遗传学方面的大量应用。 ***到目前为止，取得的进步和进步无非是引人注目。但是仍然存在许多挑战。首先，许多随机度量和过程取决于未知参数。他们在各种参数区域中的渐近行为是重要的特征，并且大多是未知的。其次，在对随机动力学的研究中，不仅有很大的需求使模型更接近现实，而且还清楚地需要对数据拟合的校准方法。需要开发更有效的统计方法。第三，数学模型（例如结合）已用于研究不断发展的人群的家谱结构。但是许多现有模型都是可交换的。一个重要的挑战将是为不可交离的家谱结构开发数学模型。最后，随机概率度量已被用作贝叶斯推断的构建基块。诸如DIRICHLET流程，破碎模型，等级迪里奇的过程，中餐厅流程和印度自助餐过程之类的模型已在统计推断中开发并广泛使用。数据积累的最新爆炸暴露了这些模型在捕获现实现象中的局限性。需要更复杂，可管理的模型。从数学上讲，人们需要开发涉及强烈局部和远距离相互作用以及复杂空间结构的随机测量模型。这项研究的潜在影响远远超出了概率理论。直接应用包括但不限于生态学中的物种抽样问题，人群遗传学中的家谱结构，物理学中的自旋玻璃模型以及金融工程中的投资组合理论。它将在学术和经济上为加拿大带来好处。 ********