Canonical decomposition of belief functions based on Teugels’ representation of the multivariate Bernoulli distribution

Abstract

A canonical decomposition of belief functions is a unique decomposition of belief functions into elementary pieces of evidence. Smets found an equivalent representation of belief functions, which he interpreted as a canonical decomposition. However, his proposal is not entirely satisfactory as it involves elementary pieces of evidence, corresponding to a generalisation of belief function axioms, whose semantics lacks formal justifications. In this paper, a new canonical decomposition relying only on well-defined concepts is proposed. In particular, it is based on a means to induce belief functions from the multivariate Bernoulli distribution and on Teugels’ representation of this distribution, which consists of the means and the central moments of the underlying Bernoulli random variables. According to our decomposition, a belief function results from as many crisp pieces of information as there are elements in its domain, and from simple probabilistic knowledge concerning their marginal reliability and the dependencies between their reliability. In addition, we show that instead of interpreting with some difficulty Smets’ representation of belief functions as a canonical decomposition, it is possible to give it a different and well-defined semantics in terms of measures of information associated with the reliability of the pieces of information in our decomposition.