Depth-Bounded Approximations of Probability

This paper introduces and investigates Depth-bounded Belief functions, a logic-based representation of quantified uncertainty. Depth-bounded Belief functions are based on the framework of Depth-bounded Boolean logics [4], which provide a hierarchy of approximations to classical logic. Similarly, Depth-bounded Belief functions give rise to a hierarchy of increasingly tighter lower and upper bounds over classical measures of uncertainty. This has the rather welcome consequence that “higher logical abilities” lead to sharper uncertainty quantification. In particular, our main results identify the conditions under which Dempster-Shafer Belief functions and probability functions can be represented as a limit of a suitable sequence of Depth-bounded Belief functions.