When observations come from time series, or when conditional independence assumptions need to be applied to simplify the structure of a joint distribution, the need of generalizing the classical results on total probability arises. Different definitions of conditional belief functions have been proposed in the past. The most common approach is to use conditional BFs induced by Dempster's
original evidence combination rule, a generalization of Bayes' rule to belief functions. Whatever the chosen definition of conditional belief function, the total probability problem for belief functions reads as follows.
Consider a set Theta and a disjoint partition Omega of Theta. Suppose a belief function bi is defined on each
element of the partition, and a (a-priori) belief function b0 is given on Omega itself.
We seek a total belief function on
the whole of Theta whose restriction to Omega coincides with the given a-priori b0, and whose conditional versions
(under Dempster's conditioning, for instance, but not necessarily) coincide with bi for each element Theta_i of the partition Omega.