We review an approach for lower bounding the supremum of stochastic processes by Pajor in 1984, based on convex volume inequalities, and its relation to some open problems in empirical process theory. We also discuss Barvinok’s thrifty approximation of convex body problem, and some of its applications in statistics. Combining these two ideas (among other things), we provide the first complete solution to an open problem of Thomas Cover 1987 about the capacity of a relay channel.
(Based on work published at Probability Theory and Related Fields, Jan. 2022, arxiv 2012.14521).
From both a statistical and practical perspective, the estimation of prediction uncertainty is a critical aspect of deep learning (DL) models. While standard deep learning models do not provide uncertainty estimates for their predictions, much recent research has focused on obtaining such estimates. Despite this, little attention has been given to the quality of estimated uncertainty from these methods. We will discuss the challenges of implementing the traditional metrics for assessing estimated uncertainty quality, interval coverage and width, for deep learning problems. Prediction interval coverage and width metrics will be given for several uncertainty-enabled deep learning models on (1) a simple regression problem and (2) a binary classification problem. We will also discuss both current and future research directions related to assessing uncertainty quality in DL models.
We propose a new solution under the Bayesian framework to simultaneously estimate mean based asynchronous changepoints in spatially correlated functional time series. Unlike previous methods that assume a shared changepoint at all spatial locations or ignore spatial correlation, our method treats changepoints as a spatial process. This allows our model to respect spatial heterogeneity and exploit spatial correlations to improve estimation. Our method is derived from the ubiquitous cumulative sum (CUSUM) statistic that dominates changepoint detection in functional time series. However, instead of directly searching for the maximum of the CUSUM based processes, we build spatially correlated two-piece linear models with appropriate variance structure to locate all changepoints at once. The proposed linear model approach increases the robustness of our method to variability in the CUSUM process, which, combined with our spatial correlation model, improves changepoint estimation near the edges. We demonstrate through extensive simulation studies that our method outperforms existing functional changepoint estimators in terms of both estimation accuracy and uncertainty quantification, under either weak and strong spatial correlation, and weak and strong change signals. Finally, we demonstrate our method using a temperature data set and a coronavirus disease 2019 (COVID-19) study.
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