In a previous post, I argued for restricting the Law of Likelihood to mutually exclusive hypotheses and claimed that this restriction would not exclude cases of genuine scientific interest because scientists don’t test hypotheses against one that are not mutually exclusive. This restriction seems natural and is sufficient to address a counterexample due to Fitelson (2007). Steel (2007) considers the same restriction but rejects it for reasons that I consider inadequate, as I explain in this post. Chandler (2013) argues for the restriction and defends it against an objection. I suggest an amendment to that defense in this post.
Fitelson criticizes the proposal to restrict the Law of Likelihood to mutually exclusive hypotheses in his (2012). One worry he expresses is that in the context of statistical model selection, it is supposed to be one of strengths of likelihoodism, as opposed to Bayesianism, that it is capable of testing nested models against each other. Thus, a likelihoodist who restricts the Law of Likelihood to mutually exclusive hypotheses thereby gives up a key support for his or her view.
I do not find this worry compelling. Giving up this purported advantage of likelihoodism seems a small price to pay in order to save the Law of Likelihood from refutation by Fitelson’s counterexample. The claim that the likelihoodist’s ability to compare nested models against each other is an advantage for likelihoodism over Bayesianism is dubious anyway. As I discussed in my previous post, it’s not clear to me that scientists actually perform such comparisons. In addition, a Bayesian can say everything that a likelihoodist can say. The primary quarrel between likelihoodists and Bayesians is over whether it makes sense to say the things that a Bayesian can say that a likelihoodist cannot. Thus, the purported fact that a likelihoodist can compare nested models against each other would not be an advantage of likelihoodism over Bayesianism even if it were true. It would be an advantage of a likelihoodist/Bayesian account of model selection over an exclusively Bayesian account of model selection. But the final problem with Fitelson’s worry is that no one who thinks that the Law of Likelihood should apply to nested models thinks that it provides a good solution to the model selection problem anyway! The likelihood ratio for the two models is ill-defined in the absence of a prior probability distribution over their simple hypothesis components, and thus is typically unavailable from a likelihoodist perspective. A standard way to address this problem is to use the likelihood ratio of the best-fitting element of one model to the best-fitting element of the other, but this comparison can never favor the smaller model and in typical problems favors the larger model with probability one. The Law of Likelihood does not vindicate the widespread preference for simpler models. Some other account is needed, such as perhaps Forster and Sober’s appeal to overfitting (2004) or an extension of Kevin Kelly’s work on Ockham’s razor to statistical problems.
Fitelson’s worry is that restricting the Law of Likelihood to mutually exclusive hypothesis “makes it too easy” to refute (†). If the Law of Likelihood provides the correct explication of evidential favoring, then the claim that it applies only to mutually exclusive hypotheses implies that either (†) is false or for any pair of hypotheses H1 and H2 that are not mutually exclusive, no data confirms one of those hypotheses more than the other. But (†) seems innocent enough, and the claim that no data confirms one hypothesis more than another if those hypotheses are mutually consistent implies that there is no such thing as a two-place confirmation relation so long as nothing confirms a tautology.
The claim that there is no such thing as a two-place confirmation relation has some plausibility, and many likelihoodists accept it. Another option is to distinguish between two different notions of evidential favoring. The Law of Likelihood explicates a “competitive” notion of evidential favoring that applies only to mutually exclusive hypotheses, whereas (†) when combined with a measure of confirmation explicates a “comparative” notion of evidential favoring that applies to any pair of hypotheses. This proposal raises questions about which measure of evidential favoring is relevant for which purposes that I will take up in a subsequent post.
 Again, the Law of Likelihood says (LL) says that datum x favors hypothesis H1 overH2 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.