Page 143 - DCAP310_INTRODUCTION_TO_ARTIFICIAL_INTELLIGENCE_AND_EXPERT_SYSTEMS

Unit 7: Probabilistic Reasoning

items of evidence. In essence, the degree of belief in a proposition depends primarily upon the Notes
number of answers (to the related questions) containing the proposition, and the subjective
probability of each answer. Also contributing are the rules of combination that reflect general
assumptions about the data.
In this formalism, a degree of belief (also referred to as a mass) is represented as a belief function
rather than a Bayesian probability distribution. Probability values are assigned to sets of
possibilities rather than single events: their appeal rests on the fact they naturally encode evidence
in favor of propositions.

Notes Dempster – Shafer theory assigns its masses to all of the non-empty subsets of the
entities that compose a system

Belief and Plausibility

Shafer’s framework allows for belief about propositions to be represented as intervals, bounded
by two values, belief (or support) and plausibility:
belief ≤ plausibility
Belief in a hypothesis is constituted by the sum of the masses of all sets enclosed by it (i.e. the
sum of the masses of all subsets of the hypothesis). It is the amount of belief that directly
supports a given hypothesis at least in part, forming a lower bound. Belief (usually denoted Bel)
measures the strength of the evidence in favor of a set of propositions. It ranges from 0 (indicating
no evidence) to 1 (denoting certainty). Plausibility is 1 minus the sum of the masses of all sets
whose intersection with the hypothesis is empty. It is an upper bound on the possibility that the
hypothesis could be true, i.e. it “could possibly be the true state of the system” up to that value,
because there is only so much evidence that contradicts that hypothesis. Plausibility (denoted by
Pl) is defined to be Pl(s)=1-Bel(~s). It also ranges from 0 to 1 and measures the extent to which
evidence in favor of ~s leaves room for belief in s. For example, suppose we have a belief of 0.5
and a plausibility of 0.8 for a proposition, say “the cat in the box is dead.” This means that we
have evidence that allows us to state strongly that the proposition is true with a confidence of
0.5. However, the evidence contrary to that hypothesis (i.e. “the cat is alive”) only has a confidence
of 0.2. The remaining mass of 0.3 (the gap between the 0.5 supporting evidence on the one hand,
and the 0.2 contrary evidence on the other) is “indeterminate,” meaning that the cat could either
be dead or alive. This interval represents the level of uncertainty based on the evidence in your
system.

Hypothesis Mass Belief Plausibility
Null (neither alive nor dead) 0 0 0
Alive 0.2 0.2 0.5
Dead 0.5 0.5 0.8
Either (alive or dead) 0.3 1.0 1.0

The null hypothesis is set to zero by definition (it corresponds to “no solution”). The orthogonal
hypotheses “Alive” and “Dead” have probabilities of 0.2 and 0.5, respectively. This could
correspond to “Live/Dead Cat Detector” signals, which have respective reliabilities of 0.2 and
0.5. Finally, the all-encompassing “Either” hypothesis (which simply acknowledges there is a
cat in the box) picks up the slack so that the sum of the masses is 1. The belief for the “Alive” and
“Dead” hypotheses matches their corresponding masses because they have no subsets; belief for
“Either” consists of the sum of all three masses (Either, Alive, and Dead) because “Alive” and

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