Level Crossing of Likelihood Functions in Sequential Decision Problems and Statistical Learning

在过去的十年里，统计方法在教育、心理学、金融和工程等不同领域的应用中取得了显著的进步。其中许多新方法的性质尚不清楚，需要有理论上的见解。这项研究的目的是(1)研究统计学习和序贯决策中有效方法的理论基础；(2)开发具有可证明的有效性和可靠性的新方法。这项研究不仅将解决统计学中的一类基本问题，而且将对其他学科的科学研究产生积极影响。一个重要的应用将是教育测试和个性化学习的适应性设计。该项目将考虑三种类型的问题：模型选择的信息量化、分类可行性的衡量和顺序分配。研究问题的一个共同特征涉及处理似然函数超过高水平的概率。对这种概率的分析在统计上是具有挑战性的，特别是在这种概率以指数速度衰减的渐近制度下。标准的数值评估工具，如蒙特卡罗方法，计算密集，以达到合理的精度水平来模拟如此小的概率。新的技术将被开发来同时获得精确的渐近逼近和被证明有效的数值方法。

The last decade has witnessed remarkable progress in statistical methods for applications arising from various fields including education, psychology, finance, and engineering. The properties of many of these new methods remain unclear, calling for theoretical insights. This research project aims at (1) studying the theoretical underpinning of effective methods in statistical learning and sequential decision making and (2) developing new methods with provably efficiency and reliability. This research will not only address a class of fundamental problems in statistics but also have a positive impact on scientific research in other disciplines. One important application will be in the adaptive design of educational testing and personalized learning. Three types of problems will be considered in the project: information quantification for model selection, measuring the feasibility of classification, and sequential allocation. A common feature of the research problems involves handling the probability that a likelihood function exceeds a high level. The analysis of such a probability is statistically challenging, especially under the asymptotic regime where this probability decays at an exponential rate. Standard numerical evaluation tools, such as Monte Carlo methods, are computationally intensive to achieve a reasonable accuracy level for simulating such a small probability. Novel techniques will be developed to obtain sharp asymptotic approximations as well as provably efficient numerical methods simultaneously.