We introduce probability calculus in this chapter as a tool for representing and reasoning with degrees of belief.

Introduction

We provide in this chapter a framework for representing and reasoning with uncertain beliefs. According to this framework, each event is assigned a degree of belief which is interpreted as a probability that quantifies the belief in that event. Our focus in this chapter is on the semantics of degrees of belief, where we discuss their properties and the methods for revising them in light of new evidence. Computational and practical considerations relating to degrees of belief are discussed at length in future chapters.

We start in Section 3.2 by introducing degrees of belief, their basic properties, and the way they can be used to quantify uncertainty. We discuss the updating of degrees of belief in Section 3.3, where we show how they can increase or decrease depending on the new evidence made available. We then turn to the notion of independence in Section 3.4, which will be fundamental when reasoning about uncertain beliefs. The properties of degrees of belief are studied further in Section 3.5, where we introduce some of the key laws for manipulating them. We finally treat the subject of soft evidence in Sections 3.6 and 3.7, where we provide some tools for updating degrees of belief in light of uncertain information.

Degrees of belief

We have seen in Chapter 2 that a propositional knowledge base Δ classifies sentences into one of three categories: sentences that are implied by Δ, sentences whose negations are implied by Δ, and all other sentences (see Figure 2.2).