Random Measures: Asymptotics, Bayesian Inference, and Stochastic Dynamics

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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.