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The general idea of belief revision is
that whenever new information becomes known, this new information
may require us to revise our beliefs. We encountered this idea
in the previous discussion of deduction and what was termed defeasible
inferences. If I believe that it is highly likely that Larry
owns a car, then it may be that when I find out that Larry lives
in Manhattan I may consider revising this belief. Or if I learn
that Larry likes carrots, then I may consider revising my belief
about Larry's ownership of a car. What does fondness for carrots
have to do with car ownership!!? The problem with deductive logic
and with standard probability theory is that there is nothing
in these formalisms that allows us to indicate that 'living in
Manhattan' and 'liking carrots' are probably not equally relevant
to questions of car ownership.
Recall
that in a probabilistic representation of knowledge, each proposition
(simple or complex) has associated with it a probability. Consequently,
the process of belief revision is one of updating the probabilities
of events when new information becomes available. The probability
p of some proposition after the receipt of information that some
proposition q has occurred is called its conditional probability.
It is written as Pr(p|q) and read as the probability of p given
that q has occurred. For example, the probability of 'a fire
in the Empire State Building' may be some small value, say .0002.
And the probability of 'smoke in the Empire State Building' may
also be some small value, say .002. But if smoke has been observed
then it may be appropriate to update or revise our estimate of
'a fire in the Empire State Building.' For example, it
might be changed to .0015.
Bayes
Rule is a formula for belief revision. This rule is given below.
Note that the propositions referred to in the formulae below
include both simple and complex propositions; and, this updating
process must be carried out for every proposition. Thus, again
we run into a problem of the tractability of this method.
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