Reasoning Under Uncertainty

1. Reasoning Under Uncertainty Kostas Kontogiannis E&CE 457

2. The Demster-Shafer Model • So far we have described techniques, all of which consider an individual hypothesis (proposition) and and assign to each of them a point estimate in terms of a CF • An alternative technique is to consider sets of propositions and assign to them an interval of the form [Belief, Plausibility] that is [Bel(p) , 1-Bel(~p)]

3. Belief and Plausibility • Belief (denoted as Bel) measures the strength of the evidence in favor of a set of hypotheses. It ranges from 0 (indicating no support) to 1 (indicating certainty). • Plausibility (denoted as Pl) is defined as Pl(s) = 1 - Bel(~s) • Plausibility also ranges from 0 to 1, and measures the extent to which evidence in favor of ~s leaves room for belief in s. In particular, if we have certain evidence in favor of ~s, then the Bel(~s) = 1, and the Pl(s) = 0. This tells us that the only possible value for Bel(s) = 0

4. Objectives for Belief and Plausibility • To define more formally Belief and Plausibility we need to start with an exhaustive universe of mutually exclusive hypotheses in our diagnostic domain. We call this set frame of discernment and we denote it as Theta • Our goal is to attach a some measure of belief to elements of Theta. In addition, since the elements of Theta are mutually exclusive, evidence in favor of some may have an effect on our belief in the others. • The key function we use to measure the belief of elements of Theta is a probability density function, which we denote as m

5. The Probability Density Function in Demster-Shafer Model • The probability density function m used in the Demster-Shafer model, is defined not just for the elements of Theta but for all subsets of it. • The quantity m(p) measures the amount of belief that is currently assigned to exactly the set p of hypotheses • If Theta contains n elements there are 2n subsets of Theta • We must assign m so that the sum of all the m values assigned to subsets of Theta is equal to 1 • Although dealing with 2n hypotheses may appear intractable, it usually turns out that many of the subsets will never need to be considered because they have no significance in a particular consultation and so their m value is 0

6. Defining Belief in Terms of Function m • Having defined m we can now define Bel(p) for a set p, as the sum of the values of m for p and for all its subsets. • Thus Bel(p) is our overall belief, that the correct answer lies somewhere in the set p • In order to be able to use m, and thus Bel and Pl in reasoning programs, we need to define functions that enable us to combine m’s that arise from multiple sources of evidence • The combination of belief functions m1 and m2 is supported by the Dempster-Shafer model and results to a new belief function m3

7. Combining Belief Functions • To combine the belief functions m1 and m2 on sets X and Y we use the following formula SumY intersect Y = Z m1(X) * m2(Y) m3(Z) = 1 - SumX intersect Y = empty m1(X) * m2(Y) • If all the intersections X, Y are not empty then m3 is computed by using only the upper part of the fraction above (I.e. normalize by dividing by 1) • If there are intersections of X, Y that are empty the upper part of the fraction is normalized by 1-k (where k is the sum of the m1*m2 on the X,Y elements that give empty intersection