分散估计中Bayes风险的信息论下界
Information-Theoretic Lower Bounds on Bayes Risk in Decentralized Estimation
摘要：
我们推导出分散估计中贝叶斯风险的下界，其中估计器没有直接存取有条件地产生的随机样本上的随机参数的兴趣，但只有从本地处理器收到的数据，观察样本。由于从处理器到估计器的通信信道中的量化和噪声，所接收的数据受到通信约束。我们首先推导出一般的贝叶斯风险的下界使用信息理论的数量，如互信息，信息密度，小球概率，和微分熵。然后，我们将这些下界应用于分散的情况下，使用强数据处理不等式来量化由于通信约束而导致的信息收缩。我们对待的情况下，一个单一的处理器和多个处理器，在不同的处理器观察到的样本可能是有条件地依赖于给定的参数，非交互式和交互式的通信协议。我们的研究结果恢复和改善最近的下界贝叶斯风险和极大极小风险的某些分散估计问题，以前只考虑条件独立的样本集和无噪声信道。此外，我们的研究结果提供了一个通用的方法来量化的估计性能下降所造成的分配资源到多个处理器，这是只讨论在现有的作品中的具体例子。
英文摘要：
We derive lower bounds on the Bayes risk in decentralized estimation, where the estimator does not have direct access to the random samples generated conditionally on the random parameter of interest, but only to the data received from local processors that observe the samples. The received data are subject to communication constraints, due to quantization and the noise in the communication channels from the processors to the estimator. We first derive general lower bounds on the Bayes risk using information-theoretic quantities, such as mutual information, information density, small ball probability, and differential entropy. We then apply these lower bounds to the decentralized case, using strong data processing inequalities to quantify the contraction of information due to communication constraints. We treat the cases of a single processor and of multiple processors, where the samples observed by different processors may be conditionally dependent given the parameter, for noninteractive and interactive communication protocols. Our results recover and improve recent lower bounds on the Bayes risk and the minimax risk for certain decentralized estimation problems, where previously only conditionally independent sample sets and noiseless channels have been considered. Moreover, our results provide a general way to quantify the degradation of estimation performance caused by distributing resources to multiple processors, which is only discussed for specific examples in existing works.
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