Well, I’ve decided to start doing philosophy blogging here again, so here is the first post since I’ve been at Princeton. I’m splitting this up into two parts, and the posts will show how to solve the problem of old evidence with the log-likelihood ratio measure. This post shows how I like to solve the traditional problem of old evidence.

Ok, for Bayesians, confirmation is defined for a hypothesis and some evidence as so :

Pr(H|E)>Pr(H).

That is, if the conditional probability of the hypothesis given the evidence is greater than the prior probability of the hypothesis, then the evidence confirms the hypothesis. But, Glymour pointed out, we have a problem when the probability of the evidence is equal to 1, for theories which deductively entail that piece of evidence. Since the theory deductively entails the evidence, Pr(E|H)=1, and we also have Pr(E)=1, so we have :

Pr(H|E)=Pr(E|H)*Pr(H)/Pr(E)=1*Pr(H)/1=Pr(H). Thus, Pr(H|E) is not greater than Pr(H), and we don’t have confirmation. The problem is that, in traditional Bayesian thought, we learn some evidence when our certainty in that evidence goes to 1. At that point, update our probabilities in the rest of what we believe by conditionalization, which proceeds as so : our new probability, Pnew, in the hypothesis is equal to Pnew(H)=Pold(H|E), where Pold(-|-) is our probability function before becoming certain of E. And this makes sense : if we become certain of E, then we should take it to be true and think what our probabilities should be for everything else given that E is true.

But, if this is so, then evidence we have already learned can never confirm any hypothesis. This is because if we have already learned the evidence, then our probability in that evidence will have already gone to 1. So, as was pointed out above, Pr(H|E) will equal Pr(H), and evidence we have already learned will not confirm any hypothesis. But this is bad, because “old evidence” – evidence we have already learned – does seem to confirm hypothesis : the old evidence of the advance of Mercury’s perihelion, for a classic example, confirmed GTR.

Several answers have been proposed, including simply using the old probability for our evidence before we learned it, or allowing Pr(E|H) to be less than 1 even when H entails E, and thus modeling confirmation by old evidence to proceed by “learning” that H deductively entails E. Neither option really suceeds. But one option readily does : ditch the strict conditionalization model provided above, and allow learning even when we do not become certain of the evidence. If this is so, then some evidence can be old, but still less than 1, and if it is then Pr(H|E)=Pr(E|H)*Pr(H)/Pr(E)=Pr(H)/Pr(E). This will always be greater than Pr(H) when Pr(E) is less than 1, so confirmation occurs. Success, the problem of old evidence goes away! Plus, there are many, many reasons to give up strict conditionalization :

It is dogmatic, for one : once your probability in some evidence goes to 1, there is no Bayesian mechanism for it to ever decrease. But surely sometimes you will have reason to become less certain of lots of ordinary, every-day evidence you were once certain of. There is no way to model this via strict conditionalization.

It is archaic, for another : it presumes an old-fashioned version of foundationalism that is no longer all that popular, even among foundationalists. However it gets spelled out, it usually requires there to be basic, perceptual beliefs that we can become absolutely certain of via our sense perceptions. But what beliefs are those?

It disrespects skepticism, for another : even if we think the skeptical challenge can be answered, there is a sense in which skeptical scenarios are live possibilities. We hope to provide reasons to think we don’t live in such a skeptical scenario, perhaps, but (few of us) think we can show that all such scenarios aren’t even possible. Now, it is tricky (and probably wrong) to equate probability 0 with “impossible.” But there is certainly a sense in which anything we assign a probability of 0 to is not a “live” possibility for us, and if we disregard problems with scenarios in which we have to decide between uncountably many possibilities, we can get away with equating probability 0 and impossibility. But if we do so, then we are counting every skeptical hypothesis as impossible. This is because if our probability in some evidence is 1, then our probability in the negation of the evidence is 0. But the negation of our evidence will be true in every (or almost every) skeptical hypothesis. So, by assigning a probability of 1 to our evidence, we are assigning a probability of 0 to every skeptical hypothesis.

And there are more such problems with requiring a probability of 1 to be applied to our evidence. We also have formal Bayesian methods (basically, Jeffrey conditionalization, but also some others) to update our beliefs when we “learn” some piece of evidence, but with less than certainty. So that seems the right path to go on independent grounds, and if we take it, then the problem of old evidence never arises, even from the start. Unfortunately, a new problem of old evidence lurks right around the corner.

Tags: confirmation theory, log-likelihood ratio measure, problem of old evidence

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