We study sequential probability assignment in the Gaussian setting, where the goal is to predict, or equivalently compress, a sequence of real-valued observations almost as well as the best Gaussian distribution with mean constrained to a given subset of $\mathbf{R}^n$. First, in the case of a convex constraint set $K$, we express the hardness of the prediction problem (the minimax regret) in terms of the intrinsic volumes of $K$; specifically, it equals the logarithm of the Wills functional from convex geometry. We then establish a comparison inequality for the Wills functional in the general nonconvex case, which underlines the metric nature of this quantity and generalizes the Slepian-Sudakov-Fernique comparison principle for the Gaussian width. Motivated by this inequality, we characterize the exact order of magnitude of the considered functional for a general nonconvex set, in terms of global covering numbers and local Gaussian widths. This implies metric isomorphic estimates for the log-Laplace transform of the intrinsic volume sequence of a convex body. As part of our analysis, we also characterize the minimax redundancy for a general constraint set. We finally relate and contrast our findings with classical asymptotic results in information theory.

翻译：我们研究高斯设定下的序贯概率赋值问题，其目标是预测（或等价地压缩）实值观测序列，其性能几乎不差于均值约束在$\mathbf{R}^n$的给定子集上的最优高斯分布。首先，对于凸约束集$K$，我们用$K$的内蕴体积表示预测问题的难度（极小极大遗憾）；具体而言，该值等于凸几何中威尔斯泛函的对数。随后，我们在一般非凸情形下建立威尔斯泛函的比较不等式，这凸显了该量的度量本质，并推广了高斯宽度的Slepian-Sudakov-Fernique比较原理。受此不等式启发，我们通过全局覆盖数与局部高斯宽度刻画了一般非凸集上该泛函的精确量级，从而为凸体的内蕴体积序列的对数拉普拉斯变换建立了度量同构估计。作为分析的一部分，我们还刻画了一般约束集的极小极大冗余。最后，我们将研究结果与信息论中的经典渐近结论进行关联与对比。