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. The Law of Likelihood is supposed to address questions about “what the evidence says about the competition between two hypotheses“ (Sober 2008, 34), but hypotheses that are not mutually exclusive are not truly in competition with one another. Thus, a natural response to this counterexample is to stipulate that the Law of Likelihood only applies to pairs of hypotheses that are mutually exclusive. There is no threat of a slightly modified version of the counterexample with mutually exclusive hypotheses: if H_{1} and H_{2} are mutually exclusive, then E cannot guarantee the truth of H_{2} if it has nonzero probability on H_{1} (assuming that nothing else guarantees the falsity of H_{1}).

One reason the Law of Likelihood is usually stated without reference to a restriction to mutually exclusive hypotheses is that statisticians typically have in mind *statistical *hypotheses to the effect that some observable random variable X has a particular (objective) probability distribution. Two such hypotheses are automatically mutually exclusive if they are distinct.[1]

The hypotheses in the example, by contrast, are *substantive* hypotheses that generate probability distributions by conditioning but are not themselves probability distributions. I see no objection to extending the Law of Likelihood to substantive as well as statistical hypotheses provided that the substantive hypotheses are mutually exclusive.

One might think that requiring mutually exclusive hypotheses is unduly restrictive because scientists often test hypotheses that are not mutually exclusive against one another. For instance, according to Machery (2013), in Greene et al. (2001), Greene and his colleagues are best understood as using a likelihoodist methodology to test the following hypotheses against one another:

H

_{1}: People respond differently to moral-personal and moral-impersonal dilemmas because the former elicit more emotional processing than the latter.H

_{2}: People respond differently to moral-personal and moral-impersonal dilemmas because the single moral rule that is applied to both kinds of dilemmas (for example, the doctrine of double effect) yields different permissibility judgments.

H_{1} and H_{2} thus stated are highly ambiguous. There are plausible ways of fleshing them out that make them mutually consistent. For instance, one could understand H_{1} as the claim that the true causal graph for the set of variables {Dilemma Type [personal, impersonal], Emotional Processing Elicited [0-10], Judgment Type [consequentialist, deontological]}, say, has arrows from Dilemma Type to Emotional Processing Elicited to Judgment Type, and similarly for H_{2}. There are other ways of fleshing them out that make them mutually exclusive. For instance, one could understand H_{1} as asserting that appealing to differences in the amount of emotional processing elicited will allow one to account for a wider range of possible experiments concerning differences in responses to moral-personal and moral-impersonal dilemmas in a more satisfactory way than appealing to moral rules, and understand H_{2} to assert the opposite.

My claim is that for any disambiguations H_{1}’ and H_{2}’ of H_{1} and H_{2}, respectively, it makes sense to talk about testing H_{1}’ against H_{2}’ only if H_{1}’ and H_{2}’ are mutually exclusive. Perhaps it would be best to think of H_{1} and H_{2} themselves not as hypotheses, but rather as merely programmatic expressions of the stances of their respective research programs; and to think of particular experiments in this domain not as testing a disambiguation of H_{1} against a disambiguation of H_{2} directly, but rather as testing a specific hypothesis in the spirit of H_{1} against an incompatible hypothesis in the spirit of H_{2}. H_{1} and H_{2} themselves are not tested against one another directly, but rather judged by their fruitfulness, empirical adequacy, and so on, in the light of many such experiments.

In any case, my claim that the Law of Likelihood does not apply to H_{1} and H_{2} themselves does not seem to be in serious conflict with Machery’s treatment of Greene et al.’s work. I would merely add to Machery’s treatment that, properly speaking, what Greene et al. in fact test against one another are not H_{1} and H_{2} themselves but rather more specific, incompatible pairs of hypotheses that are “affiliated with” H_{1} and H_{2}, respectively, in possibly various ways. I conjecture that a similar treatment will be possible whenever scientists are doing something sensible that looks like testing non-mutually-exclusive hypotheses against one another, and thus that restricting the scope of the Law of Likelihood in the way I suggest will not limit its applicability.

Unsurprisingly, I am not the first to propose restricting LL to mutually exclusive hypotheses. I will consider previous presentations of this proposal and Fitelson’s objections to it in subsequent posts.

[1] Probability density functions that differ only on sets of measure zero are mathematically distinct but “empirically compatible” in the sense that they imply all the same probabilities for observable events. Such probability density functions are not distinct in the sense that matters for statistical practice.

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