We consider Bayesian multiple hypothesis problem with independent and identically distributed observations. The classical, Sanov's theorem-based, analysis of the error probability allows one to characterize the best achievable error exponent. However, this analysis does not generalize to the case where the true distributions of the hypothesis are not exact or partially known via some nominal distributions. This problem has practical significance, because the nominal distributions may be quantized versions of the true distributions in a hardware implementation, or they may be estimates of the true distributions obtained from labeled training sequences as in statistical classification. In this paper, we develop a type-based analysis to investigate Bayesian multiple hypothesis testing problem. Our analysis allows one to explicitly calculate the error exponent of a given type and extends the classical analysis. As a generalization of the proposed method, we derive a robust test and obtain its error exponent for the case where the hypothesis distributions are not known but there exist nominal distribution that are close to true distributions in variational distance.
翻译:我们认为贝叶斯的多重假设存在独立且分布相同的观测问题。 古典, Sanov 的理论理论, 对错误概率的分析使得人们能够描述最佳可实现的错误提示。 但是, 本分析没有概括到假设的真实分布并不精确或通过某种名义分布部分为人们所知的情况。 这个问题具有实际意义, 因为名义分布可能是硬件实施中真实分布的量化版本, 或者它们可能是从标签培训序列中获取的真实分布与统计分类一样的估计数。 在本文中, 我们开发了一种基于类型的分析, 以调查巴伊西亚多重假设测试问题。 我们的分析允许一个人明确计算某一类型错误的缩写, 并扩展经典分析。 作为拟议方法的概括性, 我们得出一个强有力的测试, 并获得其错误提示, 因为假设分布并不为已知, 但是存在接近于变异距离中真实分布的标称分布 。