In the previous post, I discussed a counterexample to the Law of Likelihood due to Fitelson (2007). 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. Fitelson’s counterexample is as follows:
…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 argued for blocking this counterexample by building into the Law of Likelihood the requirement that H1 and H2 be mutually exclusive.
Steel (2007) suggests a similar maneuver in response to a variant on the “tacking paradox.” The tacking paradox has been presented as an objection to theories of confirmation which imply that E confirms H to the same degree that it confirms H conjoined with an irrelevant proposition A. The Law of Likelihood is compatible with the “contrastivist” view that there is no such thing as confirmation for a single hypothesis in isolation, so it is not subject to the tacking paradox in this form. However, it is subject to a slight variant of the paradox because it implies that E is evidentially neutral between H and H conjoined with an irrelevant proposition A.
Fitelson’s example is not an instance of the tacking paradox because neither H1 nor H2 is the conjunction of the other with an irrelevant proposition: H1 is the conjunction of H2 with itself, which is not irrelevant. But there are instances of the tacking paradox that are like Fitelson’s example in that they show that the Law of Likelihood without a restriction to mutually exclusive hypotheses violates the following intuitive restriction on accounts of evidential favoring:
(*) If E provides conclusive evidence for H1, but non-conclusive evidence for H2 (where it is assumed that E, H1, and H2 are all contingent claims), then E favors H1 over H2.
For instance, change H1 in Fitelson’s example to “the card is black and the price of tea in China was higher on January 1, 2013 than it was on January 1, 2012.“ Then (adapting Fitelson’s words above), we have P(E|H1)=Pr(E|H2)=1, but it seems absurd to claim that E is evidentially neutral between H1 and H2, as is implied by the (LL). After all, E guarantees the truth of H2 but provides only non-conclusive evidence for the truth of H1.
Steel points out that restricting the Law of Likelihood to mutually exclusive hypotheses prevents violations of (*). However, he claims that an additional restriction is needed to address fully the concern raised by the tacking paradox because one could tack an irrelevant proposition A onto one of a mutually exclusive pair of hypotheses H1 and H2. The restriction he proposes is that H1 and H2 be “structurally identical,” meaning that they assign values to the same set of random variables. That restriction prevents tacking on because A would not be irrelevant if its conjunction with one of H1 and H2 were structurally identical to H1 and H2.
Steel points out that statisticians typically work with sets of hypotheses that are mutually exclusive and structurally identical (partitions of some possibility space). However, he claims that “there are many scientifically interesting cases that do not involve the comparison of structurally identical alternatives,” such as the comparison between Newtonian mechanics and Einstein’s general theory of relativity, that are often discussed in the Bayesian confirmation literature. In addition, there are cases that involve comparing mutually consistent hypotheses. For instance, one might want to know whether the evidence supports incorporating A as part of the hypothesis H. One might for the purpose of deciding this issue consider whether E confirms better H or the conjunction of H and A.
I personally don’t find the tacking paradox paradoxical, and regard the question whether it is paradoxical or not uninteresting because it does not arise in either scientific practice or everyday life. If the strongest objection to a given theory of confirmation or evidential favoring is that it is susceptible to the tacking paradox, then I would regard that theory as a success. For that reason, I will not restrict the Law of Likelihood to structurally identical alternatives.
Let us then pass on to Steel’s claim that one might want to consider whether E confirms better H or the conjunction of H and A in order to assess whether the evidence supports incorporating A as part of the hypothesis H. What is immediately relevant is actually a slightly different claim, namely that one might want to consider whether E favors H over the conjunction of H and A and A in order to assess whether the evidence supports incorporating A as part of the hypothesis H.
A real example may help in eliciting intuitions. Let H be Darwin’s theory of evolution, A be the claim that birds are descended from dinosaurs, and E be the first archaeopteryx fossil discovered. Suppose you accept Darwin’s theory and want to assess whether the first archaeopteryx fossil discovered supports incorporating the claim that birds are descended from dinosaurs as part of that theory. Is the relevant question for this purpose whether or not the fossil favors Darwin’s theory over the conjunction of Darwin’s theory with the claim that birds are descended from dinosaurs?
I think it’s clear that the answer is “no.” If there is a relevant question here that the Law of Likelihood addresses, it is not about the “competition” between Darwin’s theory and Darwin’s theory conjoined with the claim that birds are descended from dinosaurs, but rather about the competition between Darwin’s theory conjoined with the claim that birds are descended from dinosaurs and Darwin’s theory conjoined with the negation of the claim that birds are descended from dinosaurs. If this competition is inconclusive, then one may wish to accept Darwin’s theory while remaining agnostic about the ancestry of birds.
In summary, Steel considers restricting the Law of Likelihood to mutually exclusive and structurally identical hypotheses but argues that those restrictions exclude cases of scientific interest. I see no need for the restriction to structurally identical hypotheses because I am not bothered by the tacking paradox. I do see a need for the restriction to mutually exclusive hypotheses to block counterexamples like Fitelson’s. It is fortunate, then, that I find Steel’s argument for the claim that that restriction excludes cases of genuine scientific interest unconvincing.
 A frequentist would say “parameters” rather than “random variables.”
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