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By Tarantello G.

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Applying the product rule and sum rule repeatedly, we have 210 2: Qualitative Properties 210 p(A + BjC ) = 1 and nally, p(A B jC ) = 1 p(AjC ) p(BjAC ) = 1 p(AjC )[1 p(B jAC )] = p(AjC ) + p(AB jC ) = p(AjC ) + p(B jC ) p(AjBC ) = p(AjC ) + p(B jC )[1 p(AjBC )] p(A + B jC ) = p(AjC ) + p(BjC ) p(ABjC ) : (2{48) This generalized sum rule is one of the most useful in applications. Evidently, the primitive sum rule (2{47) is a special case of (2{48), with the choice B = A. 1 Is it possible to nd a general formula for p(C jA + B), analogous to (2{48), from the product and sum rules?

As we see from this, in common language { even in mathematics textbooks { we have learned to read subtle nuances of meaning into the exact phrasing, probably without realizing it until an example like this is pointed out. We interpret \A is B " as asserting rst of all, as a kind of major premise, that A \exists"; and the rest of the statement is understood to be conditional on that premise. Put di erently, in common grammar the verb \is" implies a distinction between subject and object, which the symbol \=" does not have in formal logic or in conventional mathematics.

But now we have more than the mere qualitative statement (2{60). In Chapter 1 we wondered, without answering: What determines whether the evidence B elevates A almost to certainty, or has a negligible e ect on its plausibility? The answer from (2{54) is that, since p(B jAC ) cannot be greater than unity, a large increase in the plausibility of A can occur only when p(B jC ) is very small. Observing the gentleman's behavior (B ) makes his guilt (A) seem virtually certain, because that behavior is otherwise so very unlikely on the background information; no policeman has ever seen an innocent person behaving that way.

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A quantization property for blow up solutions of singular Liouville-type equations by Tarantello G.


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