  We develop a new framework of uncertainty variables to model uncertainty. An
uncertainty variable is characterized by an uncertainty set, in which its
realization is bound to lie, while the conditional uncertainty is characterized
by a set map, from a given realization of a variable to a set of possible
realizations of another variable. We prove Bayes' law and the law of total
probability equivalents for uncertainty variables. We define a notion of
independence, conditional independence, and pairwise independence for a
collection of uncertainty variables, and show that this new notion of
independence preserves the properties of independence defined over random
variables. We then develop a graphical model, namely Bayesian uncertainty
network, a Bayesian network equivalent defined over a collection of uncertainty
variables, and show that all the natural conditional independence properties,
expected out of a Bayesian network, hold for the Bayesian uncertainty network.
We also define the notion of point estimate, and show its relation with the
maximum a posteriori estimate. Probability theory starts with a distribution
function (equivalently a probability measure) as a primitive and builds all
other useful concepts, such as law of total probability, Bayes' law,
independence, graphical models, point estimate, on it. Our work shows that it
is perfectly possible to start with a set, instead of a distribution function,
and retain all the useful ideas needed for state estimation and inference.
