(27th-August-2020)
• Evidence e will rule out all possible worlds that are incompatible with e. Like the definition of logical consequence, the given formula e selects the possible worlds in which e is true. Evidence e induces a new measure, µe, over possible worlds where all worlds in which e is false have measure 0, and the remaining worlds are normalized so that the sum of the measures of the worlds is 1.
• The definition of the measure follows from two intuitive properties:
• If S is a set of possible worlds, all of which have e true, define µe(S) = c×µ(S) for some constant c (which we derive below).
• If S is a set of worlds, all of which have e false, define µe(S)=0.
• We want µe to be a probability measure, so if Ω is the set of all possible worlds, µe(Ω)=1. Thus, 1=µe(Ω)=µe({ω:ω e})+µe({ω:ω e})=c×µ({ω:ω e})+0=c×P(e). Therefore, c=1/P(e).
• The conditional probability of formula h given evidence e is the measure, using µe, of the possible worlds in which h is true. That is,
• P(h|e)
• = µe({ ω:ω h} )
• = µe({ ω:ω h∧e} ) + µe({ ω:ω h∧¬e} )
• = µ({ ω:ω h∧e})/P(e)+0
• = P(h∧e)/P(e).
• The last form above is usually given as the definition of conditional probability.
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