The Law of Likelihood (LL) says that datum *x* favors hypothesis *H*_{1} over *H*_{2 }if and only if the likelihood ratio *k*=Pr(*x*;*H*_{1})/Pr(*x*;*H*_{2}) 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, H

_{1}=the card is the ace of spades, and H_{2}=the card is black. In this example… P(E|H_{1})=1>Pr(E|H_{2})=1/2, but it seems absurd to claim that E favors H_{1}over H_{2}, as is implied by the (LL). After all,E guarantees the truth of H, but E provides only non-conclusive evidence for the truth of H_{2}_{1}.

I agree with Fitelson that it seems absurd to claim that E favors H_{1} over H_{2} 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…]