The steadily increasing amount of atmospheric carbon dioxide (CO$_2$) is affecting the global climate system and threatening the long-term sustainability of Earth's ecosystem. In order to better understand the sources and sinks of CO$_2$, NASA operates the Orbiting Carbon Observatory-2 & 3 satellites to monitor CO$_2$ from space. These satellites make passive radiance measurements of the sunlight reflected off the Earth's surface in different spectral bands, which are then inverted in an ill-posed inverse problem to obtain estimates of the atmospheric CO$_2$ concentration. In this work, we propose a new CO$_2$ retrieval method that uses known physical constraints on the state variables and direct inversion of the target functional of interest to construct well-calibrated frequentist confidence intervals based on convex programming. We compare the method with the current operational retrieval procedure, which uses prior knowledge in the form of probability distributions to regularize the problem. We demonstrate that the proposed intervals consistently achieve the desired frequentist coverage, while the operational uncertainties are poorly calibrated in a frequentist sense both at individual locations and over a spatial region in a realistic simulation experiment. We also study the influence of specific nuisance state variables on the length of the proposed intervals and identify certain key variables that can greatly reduce the final uncertainty given additional deterministic or probabilistic constraints, and develop a principled framework to incorporate such information into our method.
Risk quantification for the thresholding rule for multiple testing using Gaussian scale mixtures
本文研究了一类大类高斯尺度混合先验的贝叶斯多重检验的渐近性质。我们研究了两种类型的多重测试风险：Bogdan等人（2011）提出的贝叶斯风险，其中数据假设来自正常值的混合，以及与Arias-Castro和Chen提出的风险类似的频率论风险（2017）。根据van der Pas et al.（2016）的工作，我们给出了先验的一般条件，使得两个风险都有界。对于贝叶斯风险，边界几乎是尖锐的。这一结果表明，在这些条件下，所考虑的连续先验类可以与通常的两组模型（例如尖峰和板先验）竞争。我们还表明，如果非零组件的参数是足够大，极大极小风险可以渐近零。获得的分离速率与从现有文献中可以推测的分离速率一致（参见van der Pas等人，2017年b）。对于这两个问题，我们然后给出的条件下，可以得到一个自适应版本的结果。
