论文标题
通过观察到的数据缩放的贝叶斯非线性回归中的固有正则化效应
Intrinsic regularization effect in Bayesian nonlinear regression scaled by observed data
论文作者
论文摘要
Occam的剃须刀是一种指导原则，即模型应该足够简单以描述观察到的数据。虽然贝叶斯模型选择（BMS）通过固有的正则化效应（IRE）体现了它，但观察到的数据尺度尚未完全了解IRE。在有条件独立的观察结果的非线性回归中，我们表明IRE是通过观察值的细度来缩放的，这是由观察到的数据的数量和质量定义的。我们引入了一个可观察到的量化IRE的可观察到的，该IRE被称为贝叶斯特定热量，灵感来自统计推断和统计物理学之间的对应关系。我们得出了它与观察的细度相关的缩放关系。我们证明，由BMS选择的最佳模型在观察值的临界值的临界值上变化，并随着IRE的变化而变化。随着观测的细度的提高，这些变化是从选择粗粒模型到细粒的模型。当观察到的数据不足时，我们的发现扩展了对BMS典型性的理解。
Occam's razor is a guiding principle that models should be simple enough to describe observed data. While Bayesian model selection (BMS) embodies it by the intrinsic regularization effect (IRE), how observed data scale the IRE has not been fully understood. In the nonlinear regression with conditionally independent observations, we show that the IRE is scaled by observations' fineness, defined by the amount and quality of observed data. We introduce an observable that quantifies the IRE, referred to as the Bayes specific heat, inspired by the correspondence between statistical inference and statistical physics. We derive its scaling relation to observations' fineness. We demonstrate that the optimal model chosen by the BMS changes at critical values of observations' fineness, accompanying the IRE's variation. The changes are from choosing a coarse-grained model to a fine-grained one as observations' fineness increases. Our findings expand an understanding of BMS's typicality when observed data are insufficient.
