MMSE的Ziv-Zakai下界综合研究
A Comprehensive Study on Ziv-Zakai Lower Bounds on the MMSE
AI总结：
本文综合研究MMSE的Ziv-Zakai家族下界，推导通用版本并扩展至多元/M元场景，分析其紧性等性质，结果表明部分ZZ下界优于Cramér-Rao等经典贝叶斯下界。
摘要：
本文探讨了属于Ziv-Zakai（ZZ）家族的最小均方误差（MMSE）的贝叶斯下界。ZZ技术依赖于将下界与M元假设检验问题联系起来。ZZ下界（ZZB）存在三个版本：第一个依赖于所谓的填谷函数（VFF），第二个省略VFF，第三个即单点ZZB（SZZB）使用单点最大化。本文第一部分提供了这些下界的最通用版本。首先，证明这些下界在对估计量的分布没有任何假设的情况下仍然成立。其次，将SZZB扩展到M元设置，并提供了其多元情况的版本。第二部分提供了下界的一般性质。首先，证明所有下界都具有张量积性质。其次，提供了高噪声渐近的完整刻画，用于论证下界的紧性。第三，针对混合输入分布和高斯加性噪声信道提供了低噪声渐近结果。具体而言，在低噪声下，证明SZZB并不总是紧的。第三部分评估了下界的紧性。首先，证明在低噪声情况下，无VFF的ZZB对混合输入分布和高斯加性噪声信道是紧的。其次，对于离散输入，有VFF的ZZB始终是次优的，而无VFF时等于零。第三，与ZZB不同，给出了一个SZZB对离散输入MMSE紧的例子。第四，提供了下界紧性的充分必要条件。最后，展示了一些ZZ家族下界优于其他知名贝叶斯下界（即Cramér-Rao下界和最大熵下界）的例子。
英文摘要：
This paper explores Bayesian lower bounds on the minimum mean squared error (MMSE) that belong to the Ziv-Zakai (ZZ) family. The ZZ technique relies on connecting the bound to an M-ary hypothesis testing problem. Three versions of the ZZ bound (ZZB) exist: the first relies on the so-called valley-filling function (VFF), the second omits the VFF, and the third, i.e., the single-point ZZB (SZZB), uses a single point maximization. The first part of this paper provides the most general version of the bounds. First, it is shown that these bounds hold without any assumption on the distribution of the estimand. Second, the SZZB bound is extended to an M-ary setting and a version of it for the multivariate case is provided. In the second part, general properties of the bounds are provided. First, it is shown that all the bounds tensorize. Second, a complete characterization of the high-noise asymptotic is provided, which is used to argue about the tightness of the bounds. Third, the low-noise asymptotic is provided for mixed-input distributions and Gaussian additive noise channels. Specifically, in the low-noise, it is shown that the SZZB is not always tight. In the third part, the tightness of the bounds is evaluated. First, it is shown that in the low-noise regime the ZZB bound without the VFF is tight for mixed-input distributions and Gaussian additive noise channels. Second, for discrete inputs, the ZZB with the VFF is shown to be always sub-optimal, and equal to zero without the VFF. Third, unlike for the ZZB, an example is shown for which the SZZB is tight to the MMSE for discrete inputs. Fourth, sufficient and necessary conditions for the tightness of the bounds are provided. Finally, some examples are shown in which the bounds in the ZZ family outperform other well-known Bayesian bounds, i.e., the Cramér-Rao bound and the maximum entropy bound.
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