  We define a new notion of conditional belief, which plays the same role for
Dempster-Shafer belief functions as conditional probability does for
probability functions. Our definition is different from the standard definition
given by Dempster, and avoids many of the well-known problems of that
definition. Just as the conditional probability Pr (lB) is a probability
function which is the result of conditioning on B being true, so too our
conditional belief function Bel (lB) is a belief function which is the result
of conditioning on B being true. We define the conditional belief as the lower
envelope (that is, the inf) of a family of conditional probability functions,
and provide a closed form expression for it. An alternate way of understanding
our definition of conditional belief is provided by considering ideas from an
earlier paper [Fagin and Halpern, 1989], where we connect belief functions with
inner measures. In particular, we show here how to extend the definition of
conditional probability to non measurable sets, in order to get notions of
inner and outer conditional probabilities, which can be viewed as best
approximations to the true conditional probability, given our lack of
information. Our definition of conditional belief turns out to be an exact
analogue of our definition of inner conditional probability.
