This paper explores the Ziv-Zakai bound (ZZB), which is a well-known Bayesian lower bound on the Minimum Mean Squared Error (MMSE). First, it is shown that the ZZB holds without any assumption on the distribution of the estimand, that is, the estimand does not necessarily need to have a probability density function. The ZZB is then further analyzed in the high-noise and low-noise regimes and shown to always tensorize. Finally, the tightness of the ZZB is investigated under several aspects, such as the number of hypotheses and the usefulness of the valley-filling function. In particular, a sufficient and necessary condition for the tightness of the bound with continuous inputs is provided, and it is shown that the bound is never tight for discrete input distributions with a support set that does not have an accumulation point at zero.

翻译：本文探讨了Ziv-Zakai界（ZZB），这是最小均方误差（MMSE）的一个著名贝叶斯下界。首先，证明了ZZB在不对估计量的分布作任何假设的情况下成立，即估计量不一定需要具有概率密度函数。随后，在高噪声和低噪声条件下进一步分析了ZZB，并证明了它始终具有张量积性质。最后，从多个方面研究了ZZB的紧致性，包括假设的数量以及谷值填充函数的效用。特别地，本文给出了连续输入下该界紧致性的充分必要条件，并证明了对于支撑集在零点无聚点的离散输入分布，该界永远不紧致。