(24th-August-2020)
• Probability theory is built on the same foundation of worlds and variables as constraint satisfaction (see Section 4.2). Instead of having constraints that eliminate some worlds and treat every other world as possible, probabilities put a measure over the possible worlds. The variables in probability theory are referred to as random variables. The term random variable is somewhat of a misnomer because it is neither random nor variable. As discussed in Section 4.2, worlds can be described in terms of variables; a world corresponds to an assignment of a value to each variable. Alternatively, variables can be described in terms of worlds; a variable is a function that returns a value on each world.
• First we define probability as a measure on sets of worlds, then define probabilities on propositions, then on variables.
• A probability measure over the worlds is a function µ from sets of worlds into the non-negative real numbers such that
• if Ω1 and Ω2 are disjoint sets of worlds (i.e., if Ω1∩Ω2={}), then µ(Ω1∪Ω2)=µ(Ω1)+µ(Ω2);
• if Ω is the set of all worlds, µ(Ω)=1.
• For example, the worlds correspond to the possible real-valued heights, in centimeters, of a particular person. In this example, there are infinitely many possible worlds. The measure of the set of heights in the range [175,180) could be 0.2 and the measure of the range [180,190) could be 0.3. Then the measure of the range [175,190) is 0.5. However, the measure of any particular height could be zero.
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• The probability of proposition α, written P(α), is the measure of the set of possible worlds in which α is true. That is,
• P(α)=µ({ω: ω α} ),
• where ω α means α is true in world ω. Thus, P(α) is the measure of the set of worlds in which α is true.
• This use of the symbol differs from its use in the previous chapter (see Section 5.1). There, the left-hand side was a knowledge base; here, the left-hand side is a world. Which meaning is intended should be clear from the context.
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