A New Approach to Updating Beliefs

TL;DR

This paper introduces a novel definition of conditional belief for Dempster-Shafer belief functions, aligning it with the concept of conditional probability, and addresses limitations of previous definitions by providing a closed-form expression and extending to non-measurable sets.

Contribution

It proposes a new, well-founded approach to defining conditional belief functions that parallels conditional probability and extends to non-measurable sets, improving theoretical understanding.

Findings

Provides a closed-form expression for the new conditional belief

Connects belief functions with inner measures and conditional probabilities

Extends the concept of conditional probability to non-measurable sets

Abstract

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.