We present a new uncertainty principle for risk-aware statistical estimation, effectively quantifying the inherent trade-off between mean squared error ($\mse$) and risk, the latter measured by the associated average predictive squared error variance ($\sev$), for every admissible estimator of choice. Our uncertainty principle has a familiar form and resembles fundamental and classical results arising in several other areas, such as the Heisenberg principle in statistical and quantum mechanics, and the Gabor limit (time-scale trade-offs) in harmonic analysis. In particular, we prove that, provided a joint generative model of states and observables, the product between $\mse$ and $\sev$ is bounded from below by a computable model-dependent constant, which is explicitly related to the Pareto frontier of a recently studied $\sev$-constrained minimum $\mse$ (MMSE) estimation problem. Further, we show that the aforementioned constant is inherently connected to an intuitive new and rigorously topologically grounded statistical measure of distribution skewness in multiple dimensions, consistent with Pearson's moment coefficient of skewness for variables on the line. Our results are also illustrated via numerical simulations.
翻译:我们为风险意识统计估算提出了一个新的不确定性原则,有效地量化了平均正方差(mse$)和风险之间的内在权衡,而风险则以相应的平均预测平差差差(sev$)来衡量。我们的不确定性原则具有一种熟悉的形式,类似于其他一些领域产生的基本和经典结果,如海森堡在统计和量子力学方面的原则,加博尔限度(时间尺度权衡)在口音分析方面。特别是,我们证明,如果有国家和可观测数据的联合基因模型,则美元和美元之间的产品与以下各值之间由一个基于可折算模型的常数(sev$)约束,这显然与最近研究过的帕雷托边界(美元)统计和量子力(MMSE)估算问题中受限制的最低美元(MMSE)问题有关。此外,我们证明,上述常数与一个直观和严格基于表面的统计尺度测量多维度分布度的统计模型,也与Pearson的模型常数相联,这与我们模拟时的数值变量也与Searson的数值系数一致。