Ok, so we now know how to solve the problem of old evidence as it is traditionally stated : just refuse to allow the probability of our evidence to go to 1. But, then Earman comes along and points out a quantitative problem of old evidence. Sure, sure, he says, doing what I just recommended works. But now consider old evidence for which we are *quite *certain about, even if we don’t assign it a probability of 1. Lets say our probability in the evidence is equal to .999. Then confirmation goes like this, again for a hypothesis that entails the evidence.

Pr(H|E)=Pr(E|H)*Pr(H)/Pr(E)=1*Pr(H)/.999. This is going to be very nearly Pr(H). And so we might think that the evidence *barely *confirms the hypothesis. This will be bad, because we do have cases of old evidence where we were quite confident in that old evidence, and where that old evidence *strongly *confirmed the hypothesis (again, this is taken to be the case for Mercury and GTR). And, on some measures of confirmation – measures that define the *degree *of confirmation of a hypothesis by some evidence – this will be true. So now we have a quantitative problem of old evidence : how can old evidence that we are quite sure about strongly confirm a hypothesis?

Here are three of the most plausible measures of confirmation:

The difference measure : c(H,E)=Pr(H|E)-Pr(H)

The ratio measure : c(H,E)=log(Pr(H|E)/Pr(H))

The likelihood ratio measure c(H,E)=log(Pr(E|H)/Pr(E|~H)).

(The ratio and likelihood ratio measure are given in terms of logs in order to normalize them such that they give a value greater than 0 just in case Pr(H|E)>Pr(H)).

So, lets see what happens for them when the probability of the evidence is high (=.999) and the hypothesis entails the evidence

difference measure = Pr(H|E)-Pr(H) = Pr(E|H)*Pr(H)/Pr(E)-Pr(H)=Pr(H)/.999-Pr(H)=Pr(H)(1/.999-1)=Pr(H)*(.001001…). This value, obviously, will be quite low, and will not count on “strongly confirmed” on any satisfactory definition of “strongly confirmed” in terms of the difference measure.

ratio measure = log(Pr(H|E)/Pr(H)) = log(Pr(E|H)/Pr(E)) = log(1/.999). This value will be exceedingly small no matter your base for the logarithm, and will not count as “strongly confirmed” on any satisfactory definition of “strongly confirmed” in terms of the difference measure.

likelihood ratio measure = log(Pr(E|H)/Pr(E|~H))=log(1/Pr(E|~H)=log(1/[Pr(~H|E)*Pr(E)/Pr(~H)])=log(Pr(~H)/[Pr(~H|E)*.999]). This value can be as high as we like, as long as Pr(~H) is sufficiently larger than Pr(~H|E). In other words, as long as the old evidence we are quite certain about strongly *disconfirms *the *negation *of the hypothesis under consideration, then we will get strong confirmation from the log-likelihood ratio measure, even for very certain evidence.

And a little reflection shows us this is more or less what happens in the case of the perihelion advance of Mercury for GTR, and is exactly the condition when old evidence should strongly confirm a hypothesis. Take the second point first :

We obviously don’t want any highly certain piece of old evidence to strongly confirm any hypothesis that predicts it. Lets say we are highly certain that the sun has appeared to rise and set at certain times in the past. Now lets say we are considering a bold new hypothesis that the earth is, in fact, flat and rests on the back of a turtle swimming in an ocean, and two other turtles are swimming along on each side of him and lazily playing catch with the sun by tossing it back and forth at specified intervals. We obviously don’t want our highly certain evidence of the sun appearing to rise and set to confirm this theory, even though this theory.

Now consider a relatively more plausible case. GTR, we know, “reduces” to classical physics in low-energy cases in the sense that it recovers Newton’s predictions. But we don’t say, and don’t want to say, that GTR’s correct prediction of, say, the orbit of Mars strongly confirms GTR, because we have another theory (Newton’s) that also predicts this. In other words, if we have an alternative theory that accords with our old evidence, then that old evidence doesn’t strongly confirm our new theory. And this is exactly what the log-likelihood ratio measure says. The negation of H (here, H is GTR) can be thought of as a disjunction of each member of a partition of theories (say, our theories of gravity). Newton’s theory will be a member of that partition. So, ~H will entail E (when E is something like the orbit of Jupiter), so Pr(~H|E) won’t be “much smaller” than Pr(~H). So you won’t get strong confirmation according to the likelihood ratio measure. The likelihood ratio measure only provides strong confirmation for highly certain old evidence when there is not an alternative theory we have which also accounts for that evidence, which is exactly what we want. I’m glossing over some important questions and details here, but whatever. The likelihood ratio measure seems to provide strong confirmation in exactly the scenarios we want it to : when there isn’t an alternative theory that also predicts that old evidence.

And this, for instance, in the scenario we find ourselves in with the famous advance of the perihelion of Mercury. Newton’s classical mechanics and gravitational theory was unable to correctly predict this advance. The observation of the advance was pretty well known about prior to Einstein’s formulation of GTR. So we can say that the old evidence strongly disconfirmed classical physics (in that Pr(Classical|Perihelion) was relatively low), but the probability of classical physics was relatively high (it was strongly confirmed in many other areas). So we have a case where, plausibly, Pr(~H) was a good deal higher than Pr(~H|E), which gives us strong confirmation for GTR by the advance of the Perihelion of Mercury by the likelihood ratio measure, just as we want (and, I’m emphasizing again, I’m ignoring important questions and details, such as that classical physics + GTR don’t form a partition).

So, the ikelihood ratio measure solves the quantitative problem of old evidence. It provides strong confirmation just in those cases we want it to : when we have a relatively well-confirmed alternative theory that is inconsistent with some piece of old evidence, and that our new theory correctly predicts. And it allows us to see how GTR was strongly confirmed by the advance of the perihelion of mercury. So using the likelihood ratio measure as our measure of confirmation solves the quantiative problem of old evidence, and Bayesian confirmation theory is safe from the attacks of Glymour and Earman.

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

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