Fusing probabilistic information is a fundamental task in signal and data processing with relevance to many fields of technology and science. In this work, we investigate the fusion of multiple probability density functions (pdfs) of a continuous random variable or vector. Although the case of continuous random variables and the problem of pdf fusion frequently arise in multisensor signal processing, statistical inference, and machine learning, a universally accepted method for pdf fusion does not exist. The diversity of approaches, perspectives, and solutions related to pdf fusion motivates a unified presentation of the theory and methodology of the field. We discuss three different approaches to fusing pdfs. In the axiomatic approach, the fusion rule is defined indirectly by a set of properties (axioms). In the optimization approach, it is the result of minimizing an objective function that involves an information-theoretic divergence or a distance measure. In the supra-Bayesian approach, the fusion center interprets the pdfs to be fused as random observations. Our work is partly a survey, reviewing in a structured and coherent fashion many of the concepts and methods that have been developed in the literature. In addition, we present new results for each of the three approaches. Our original contributions include new fusion rules, axioms, and axiomatic and optimization-based characterizations; a new formulation of supra-Bayesian fusion in terms of finite-dimensional parametrizations; and a study of supra-Bayesian fusion of posterior pdfs for linear Gaussian models.

中文翻译：

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概率密度函数的融合


融合概率信息是信号和数据处理中的一项基本任务，与许多技术和科学领域相关。在这项工作中，我们研究连续随机变量或向量的多个概率密度函数 (pdf) 的融合。尽管连续随机变量的情况和pdf融合问题在多传感器信号处理、统计推断和机器学习中经常出现，但普遍接受的pdf融合方法并不存在。与 pdf 融合相关的方法、观点和解决方案的多样性促进了该领域理论和方法的统一呈现。我们讨论融合 pdf 的三种不同方法。在公理化方法中，融合规则由一组属性（公理）间接定义。在优化方法中，它是最小化涉及信息论散度或距离度量的目标函数的结果。在超贝叶斯方法中，融合中心将要融合的 pdf 解释为随机观测值。我们的工作部分是一项调查，以结构化和连贯的方式回顾文献中提出的许多概念和方法。此外，我们还为这三种方法中的每一种提供了新的结果。我们最初的贡献包括新的融合规则、公理、公理和基于优化的表征；有限维参数化的超贝叶斯融合的新表述；以及线性高斯模型后验概率密度函数的超贝叶斯融合研究。