The Law of Likelihood (LL) says that datum x favors hypothesis H1 over H2 if and only if the likelihood ratio k=Pr(x;H1)/Pr(x;H2) is greater than 1, with k measuring the degree of favoring. Fitelson (2007) offers the following as a counterexample to the Law of Likelihood:
…we’re going to draw a single card from a standard (well-shuffled) deck…. E=the card is a spade, H1=the card is the ace of spades, and H2=the card is black. In this example… P(E|H1)=1>Pr(E|H2)=1/2, but it seems absurd to claim that E favors H1 over H2, as is implied by the (LL). After all, E guarantees the truth of H2, but E provides only non-conclusive evidence for the truth of H1.
I agree with Fitelson that it seems absurd to claim that E favors H1 over H2 in this case. However, it also seems odd to speak in any way of evidence favoring one hypothesis over another when those hypotheses are not mutually exclusive. [Read more…]