Bayesian Nonparametric Methods for Aggregated and Multivariate Outputs

该项目调查了两种情况下的数据稀缺的标签数据是昂贵的，这是在许多环境和社会科学问题中出现的问题的情况。我们的目标是开发新的方法，解决这些情况下使用灵活的代理模型编码先验的信念和可解释的不确定性量化。该项目福尔斯属于EPSRC数学科学研究领域，部分由Cervest Limited（一家专注于地球科学AI的人工智能初创公司）和帝国理工学院伦敦资助并与之合作。工业界和学术界之间的这种合作将使我们的研究能够从工业界获得广泛的地球观测数据集，并使工业界能够为自己的工作获得新的方法。第一部分涉及总产出，解决了我们通常观察到或必须对大量个人或地理区域的数量进行平均的情况。出现这种问题的一个重要应用是计算药物或政策干预的平均治疗效果。当标记数据稀缺时，这类问题甚至更加复杂。例如，当我们只知道整个地区的产量时，我们如何在一个大的地理区域内建立作物产量模型？该项目的第二部分涉及模拟多个量，如降水和温度，共同利用它们的相互依赖性。同样，当标记数据稀缺时，建模多个量可以允许提取额外的信号。为了捕捉协变量和输出之间的复杂相互作用，非参数方法，假设无限多个模型参数，如高斯过程（GP），提供了一种灵活的方式来编码先验信念，并且还有丰富的文献使用GP用于标签稀缺和特征丰富的情况（Law et al.（2018）; Hamelijnck et al.（2019））。全科医生使用正态分布对先验信念进行编码，并且还可以给出不确定性量化，这在重要的情况下非常理想。最近，基于树的模型（奇普曼et al.（2010）; Lakshminarayanan et al.（2016）），其中先验信念被分解为个体或子区域的子组，已经引起了机器学习社区的兴趣，产生了与GP高度竞争的结果。像GP一样，基于树的模型也提供了一个灵活的非参数模型，可以提供不确定性量化，但基于树的先验的属性尚未被充分利用，用于更复杂的应用。我们希望致力于开发新的非参数方法，作为我们项目目标的解决方案。我们将首先开发新的非参数建模方法的应用程序，涉及汇总数量的兴趣和输出。然后，我们将致力于开发灵活的多输出模型，并考虑到环境科学的广泛应用。参考文献：奇普曼，H.A.，乔治和麦卡洛克，R.E.，2010. BART：Bayesian Additive Regression Trees贝叶斯回归树。应用统计年鉴，4（1），第266 - 298页。Hamelijnck，O.，Damoulas，T.，Wang，K.和Girolami，M.，2019.多分辨率多任务高斯过程。在神经信息处理系统的进展（pp。14025 - 14035）。Lakshminarayanan，B.，Roy，D. M.和Teh，Y.W.，2016年5月不确定性重要时的大规模回归的蒙德里安森林。在人工智能和统计学（pp。1478 - 1487）。法律，H.C.，Sejdinovic，D.，卡梅隆，E.，卢卡斯，T.，Flaxman，S.，战斗，K。和umizu，K.，2018.基于高斯过程的聚合输出的变分学习。在神经信息处理系统的进展（pp. 6081 - 6091）。

This project investigates 2 types of situations under data-scarce labelled data that are expensive to obtain, situations which are problems that occur in many environmental and social sciences problems. We aim to develop novel methods that tackle these situations using flexible proxy models that encode prior beliefs and interpretable uncertainty quantifications. This project falls within the EPSRC Mathematical Sciences research area and is partly funded by and in collaboration with Cervest Limited, an artificial intelligence start-up focusing on Earth Science AI, and Imperial College London. This collaboration between industry and academia will allow our research to have access to a wide array of Earth observation datasets from the industry as well as for the industry to gain access to novel methodologies for their own work. The first part involving aggregated outputs address the situation where we typically observe or must average out quantities over large groups of individuals or geographical areas. An important application where this type of problem occurs is in computing the average treatment effect of administering a pharmaceutical or policy intervention. When labelled data is scarce, this type of problem is even more complex. For instance, how do we model crop yields over a large geographical region when we only know what the yield is for the entire region? The second part of the project involves modelling multiple quantities, such as precipitation and temperature, jointly in a way that exploits their inter-dependence. Again, when labelled data is scarce modelling multiple quantities can allow for additional signals to be extracted. To capture complex interactions between covariates and outputs, nonparametric methods, ones that assume infinitely many model parameters such as Gaussian processes (GP), provide a flexible way for encoding prior beliefs, and there is also a rich literature on using GPs for label-scarce and feature-rich situations (Law et al. (2018); Hamelijnck et al. (2019)). GPs encode prior beliefs using normal distributions and can also give uncertainty quantification, which is highly desirable for situations when this is important. Recently, tree-based models (Chipman et al. (2010); Lakshminarayanan et al. (2016)), where the prior belief is broken down into subgroups of individuals or subregions, have been of interest to the machine learning community, yielding highly competitive results to GPs. Like GPs, tree-based models also provide a flexible nonparametric model that can provide uncertainty quantification, but properties of tree-based priors have yet to have been fully exploited for more complex applications. We hope to work on the development of novel nonparametric methodologies as solutions for our project aims. We will first develop novel nonparametric modelling approaches for applications that involve aggregated quantities of interest and outputs. We will then work on developing flexible models for multiple outputs with broad applications for environmental sciences in mind. References:Chipman, H.A., George, E.I. and McCulloch, R.E., 2010. BART: Bayesian additive regression trees. The Annals of Applied Statistics, 4(1), pp.266-298. Hamelijnck, O., Damoulas, T., Wang, K. and Girolami, M., 2019. Multi-resolution multi-task Gaussian processes. In Advances in Neural Information Processing Systems (pp. 14025-14035). Lakshminarayanan, B., Roy, D.M. and Teh, Y.W., 2016, May. Mondrian forests for large-scale regression when uncertainty matters. In Artificial Intelligence and Statistics (pp. 1478-1487). Law, H.C., Sejdinovic, D., Cameron, E., Lucas, T., Flaxman, S., Battle, K. and Fukumizu, K., 2018. Variational learning on aggregate outputs with Gaussian processes. In Advances in Neural Information Processing Systems (pp. 6081-6091).