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首页> 外文期刊>IEEE Transactions on Information Theory >Anonymous Heterogeneous Distributed Detection: Optimal Decision Rules, Error Exponents, and the Price of Anonymity
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Anonymous Heterogeneous Distributed Detection: Optimal Decision Rules, Error Exponents, and the Price of Anonymity

机译：匿名异构分布检测：最优决策规则，错误指数和匿名价格

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摘要

We explore the fundamental limits of heterogeneous distributed detection in an anonymous sensor network with n sensors and a single fusion center. The fusion center collects the single observation from each of the n sensors to detect a binary parameter. The sensors are clustered into multiple groups, and different groups follow different distributions under a given hypothesis. The key challenge for the fusion center is the anonymity of sensors-although it knows the exact number of sensors and the distribution of observations in each group, it does not know which group each sensor belongs to. It is hence natural to consider it as a composite hypothesis testing problem. First, we propose an optimal test called mixture likelihood ratio test, which is a randomized threshold test based on the ratio of the uniform mixture of all the possible distributions under one hypothesis to that under the other hypothesis. Optimality is shown by first arguing that there exists an optimal test that is symmetric, that is, it does not depend on the order of observations across the sensors, and then proving that the mixture likelihood ratio test is optimal among all symmetric tests. Second, we focus on the Neyman-Pearson setting and characterize the error exponent of the worst-case type-II error probability as n tends to infinity, assuming the number of sensors in each group is proportional to n. Finally, we generalize our result to find the collection of all achievable type-I and type-II error exponents, showing that the boundary of the region can be obtained by solving an optimization problem. Our results elucidate the price of anonymity in heterogeneous distributed detection, and can be extended to M-ary hypothesis testing with heterogeneous observations generated according to hidden latent variables.
机译：我们探索具有n个传感器和一个融合中心的匿名传感器网络中异构分布检测的基本限制。融合中心从n个传感器中的每一个收集单个观测值，以检测二进制参数。传感器分为多个组，在给定的假设下，不同的组遵循不同的分布。融合中心面临的主要挑战是传感器的匿名性，尽管它知道传感器的确切数量和每个组中观测值的分布，但它不知道每个传感器属于哪个组。因此，很自然地将其视为一个综合假设检验问题。首先，我们提出一种称为混合似然比检验的最优检验，这是一个随机阈值检验，它基于一个假设下所有可能分布的均匀混合与另一个假设下的均匀混合的比率。通过首先争论是否存在一个对称的最优​​检验来表明最优性，也就是说，它不依赖于跨传感器的观察顺序，然后证明混合似然比检验在所有对称检验中都是最优的。其次，我们集中在Neyman-Pearson设置上，并假设n组中传感器的数量与n成正比，当n趋于无穷大时，将最坏情况的II型错误概率的误差指数特征化。最后，我们对结果进行概括，以找到所有可实现的I型和II型误差指数的集合，这表明可以通过解决优化问题来获得区域的边界。我们的研究结果阐明了异构分布式检测中匿名性的代价，并且可以扩展为使用根据隐性潜在变量生成的异构观测值进行的M元假设检验。

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