Abstract
In recent years, we have studied information properties of various types
of mixtures of probability distributions and introduced a new type, which includes
previously known mixtures as special cases. These studies are disseminated in dierent
fields: reliability engineering, econometrics, operations research, probability, the information
theory, and data mining. This paper presents a holistic view of these studies
and provides further insights and examples. We note that the insightful probabilistic
formulation of the mixing parameters stipulated by Behboodian (1972) is required for
a representation of the well-known information measure of the arithmetic mixture.
Applications of this information measure presented in this paper include lifetime
modeling, system reliability, measuring uncertainty and disagreement of forecasters,
probability modeling with partial information, and information loss of kernel estimation.
Probabilistic formulations of the mixing weights for various types of mixtures provide
the Bayes-Fisher information and the Bayes risk of the mean residual function.
