A Mass-Shifting Phenomenon of Truncated Multivariate Normal Priors

We show that lower-dimensional marginal densities of dependent zero-mean normal distributions truncated to the positive orthant exhibit a <i>mass-shifting</i> phenomenon. Despite the truncated multivariate normal density having a mode at the origin, the marginal density assigns increasingly small mass near the origin as the dimension increases. The phenomenon accentuates with stronger correlation between the random variables. This surprising behavior has serious implications toward Bayesian constrained estimation and inference, where the prior, in addition to having a full support, is required to assign a substantial probability near the origin to capture flat parts of the true function of interest. A precise quantification of the mass-shifting phenomenon for both the prior and the posterior, characterizing the role of the dimension as well as the dependence, is provided under a variety of correlation structures. Without further modification, we show that truncated normal priors are not suitable for modeling flat regions and propose a novel alternative strategy based on shrinking the coordinates using a multiplicative scale parameter. The proposed shrinkage prior is shown to achieve optimal posterior contraction around true functions with potentially flat regions. Synthetic and real data studies demonstrate how the modification guards against the mass shifting phenomenon while retaining computational efficiency. Supplementary materials for this article are available online.

我们证明，截断至正卦限的相依零均值正态分布的低维边缘密度，呈现出质量偏移（mass-shifting）现象。尽管截断多元正态密度在原点处存在众数，但随着维度增加，边缘密度在原点附近分配的质量会逐渐减小。该现象会随着随机变量间相关性的增强而愈发显著。这一令人意外的特性对贝叶斯约束估计与推断具有重要影响：在此类场景中，先验除需具备全支撑外，还需在原点附近分配充足概率，以捕捉目标真实函数的平坦区域。本文针对多种相关结构，对先验与后验的质量偏移现象进行了精准量化，刻画了维度与相关性各自的作用。若不进行额外修正，截断正态先验并不适用于平坦区域的建模；为此我们提出了一种基于乘法尺度参数收缩坐标的全新替代策略。实验证明，所提出的收缩先验可在存在潜在平坦区域的真实函数附近实现最优的后验收缩。人工合成数据集与真实数据集实验验证了，该修正方法既能规避质量偏移现象，又能保持计算效率。本文的补充材料可在线获取。