Here’s a problematic line of thought about the main competitors in confirmation theory and Bayesianism in particular, arising from Branden Fitelson’s nice paper here. Consider the following theories of confirmation. First, the law of likelihood preferred by likelihoodists such as Sober:
(LL) Evidence E favors hypothesis H1 over hypothesis H2 if and only if H1 confers greater probability on E than H2 does.
Second, he weak law of likelihood:
(WLL) Evidence E favors hypothesis H1 over hypothesis H2 if Pr(E |H1) > Pr(E |H2) and Pr(E |~H1) ≤ Pr(E |~H2).
Third, an old-fashioned view:
(‡) E favors H1 over H2 if and only if Pr(H1 |E) > Pr(H2 |E).
We can eliminate the third one quickly.
About it, Fitelson says,
But, (‡) is an inadequate Bayesian theory of favoring, because the underlying notion of confirmation on which it is based ignores probabilistic relevance. Consider any case in which E raises the probability of H1, but E lowers the probability of H2. Intuitively, this is a case in which E indicates that H1 is true, but E indicates that H2 is false. In such a case, it seems obvious that E provides better evidence for the truth of H1 than for the truth of H2. And, as a result, it seems clear that, in such cases, we should say that E favors H1 over H2. Obviously, any contemporary, relevance-based Bayesian theory of favoring will have this consequence. . . . Unfortunately, (‡) does not have this consequence. In fact, according to (‡), E can favor H2 over H1 in such cases, which is absurd. For this reason, nobody defends (‡) anymore.
Here’s a concrete example Branden uses. Where we sample randomly from the natural numbers between 1 and 10 inclusive, let E be the disjunction of 1,2,8,9,10, H1 be the disjunction of 1,2 and H2 be the disjunction of 2,3,4,5,6,7,8,9. Here Pr(H1/E) = 2/5, compared to Pr(H1)=1/5; Pr(H2/E)=3/5, compared to Pr(H2)=4/5.
So the issues now surround (LL) and (WLL).
For contemporary Bayesians, confirmation is a matter of probabilistic relevance. Thus, degree of confirmation is measured using some relevance measure c(H,E) of the “degree to which E raises the probability of H”. Here are the three most popular Bayesian relevance measures of non-relational confirmation:
Difference: d(H,E) =df Pr(H |E) − Pr(H)
Ratio: r(H,E) =df Pr(H |E)/Pr(H)
Likelihood-Ratio: l(H,E) =df Pr(E |H)/Pr(E |~H)
The general form of the Bayesian account of favoring, using any of these measures, is:
Evidence E favors hypothesis H1 over hypothesis H2, according to measure c, if and only if c(H1,E) > c(H2,E).
One other point to note before getting to my point(!). Jim Joyce has shown here that all these Bayesian confirmation measures commit their defenders to (WLL), thereby arguing that (WLL) is something like a core Bayesian commitment.
Given all this, consider two of Branden’s examples.
Let E be an ace, H1 is heart ace, and H2 is club or spade ace. It is clear in this example that E favors H2 over H1, but the likelihoods do not sustain this answer, since the probability of E on either H1 or H2 is 1. So (LL) is false. But note as well that Pr(E |~H2) = 4/50 > Pr(E |~H1) = 4/51, so (WLL) does not succumb to the example.
Consider a second example. E = the card is a spade, H1 = the card is the ace of spades, and H2 = the card is black. In this example, Pr(E |H1) = 1 > Pr(E |H2) = 1/2 , but it seems absurd to claim that E favors H1 over H2, since the evidence entails H2 but not H1. Note here that not only (LL) succumbs, but (WLL) as well.
Fitelson shows that the ratio version of Bayesianism is equivalent to (LL), so it is undermined by the first example. That leaves the difference and likelihood-ratio versions. The value of d in the first example is 1/4 – 1/52 for H1 and 1/2 – 1/26 for H2, leaving us to conclude that E confirms H2 more than H1, just as we should see. In the second example, the value of d for H1 is 1/13 – 1/52, and for H2 is 1 – 1/2, showing that E favors H2 over H1.
For the value for l, both examples cause problems, since the denominator value is zero in both cases, leaving the value for l undefined in both examples, and thus leaving it not true that the evidence favors either hypothesis over the other. Perhaps this problem could be solved by abandoning a Kolmogorov treatment of probability in favor of a Réyni-Popper theory. Even then, however, the second example will create trouble.
As a result, the two examples show more than that (LL) is false, undermining likelihoodism. They also show that (WLL) is false, and thus given Joyce’s result that (WLL) is the core commitment of Bayesianism, Bayesianism is in trouble, in a way independent of the usual complaints about Bayesian reliance on priors. How bad a conclusion is this?