Random Measures: Asymptotics, Bayesian Inference, and Stochastic Dynamics

随机度量是一种度量值随机元素。随机度量的集合成为一个度量值过程。在过去的三十年里，随机测量和测值过程一直是概率论、随机过程和统计学中非常活跃的研究课题。一些研究较好的模型包括随机分配模型、合并模型、Dirichlet过程模型、棍棒断裂模型、Fleming-Viot过程模型和各种骨灰盒模型。他们研究的一个主要动机来自于对复杂的相互作用系统和相应的演变动力学进行建模。在这一领域的深入研究导致了概率论和随机过程的许多理论进展，并在天体物理、化学、通信、生态学、经济学、语言学、机器学习和种群遗传学中得到了大量的应用。 到目前为止取得的进展和进展丝毫不亚于显著。但许多挑战依然存在。首先，许多随机测量和过程依赖于未知参数。它们在各种参数区域中的渐近行为是重要的特征，而且大多未知。其次，在随机动力学的研究中，不仅有使模型更接近实际的强烈要求，而且明确需要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，如聚合体，已被用于研究进化种群的谱系结构。但许多现有的车型是可以更换的。一个重要的挑战将是为不可交换的谱系结构开发数学模型。最后，随机概率度量被用作贝叶斯推理的构建块。Dirichlet过程、棒断模型、分层Dirichlet过程、中餐馆过程和印度自助餐过程等模型已被开发并广泛应用于统计推断。最近数据积累的爆炸性增长暴露了这些模型在捕捉真实世界现象方面的局限性。需要更复杂、更易管理的模型。从数学上讲，人们需要开发随机测量模型，这些模型涉及到强烈的局部和远程相互作用，以及复杂的空间结构。 拟议的研究将通过发展新的随机测量和随机过程模型，设计有效的算法，以及分析各种复杂系统在不同极限制度下的渐近行为来解决这些挑战。这项研究的潜在影响远远超出了概率论。直接应用包括但不限于，生态学中的物种抽样问题，种群遗传学中的谱系结构，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。这将给加拿大带来学术和经济上的好处。

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.