  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.
