  A random set is a generalisation of a random variable, i.e. a set-valued
random variable. The random set theory allows a unification of other
uncertainty descriptions such as interval variable, mass belief function in
Dempster-Shafer theory of evidence, possibility theory, and set of probability
distributions. The aim of this work is to develop a non-deterministic inference
framework, including theory, approximation and sampling method, that deals with
the inverse problems in which uncertainty is represented using random sets. The
proposed inference method yields the posterior random set based on the
intersection of the prior and the measurement induced random sets. That
inference method is an extension of Dempster's rule of combination, and a
generalisation of Bayesian inference as well. A direct evaluation of the
posterior random set might be impractical. We approximate the posterior random
set by a random discrete set whose domain is the set of samples generated using
a proposed probability distribution. We use the capacity transform density
function of the posterior random set for this proposed distribution. This
function has a special property: it is the posterior density function yielded
by Bayesian inference of the capacity transform density function of the prior
random set. The samples of such proposed probability distribution can be
directly obtained using the methods developed in the Bayesian inference
framework. With this approximation method, the evaluation of the posterior
random set becomes tractable.
