Wednesday, October 31, 2007

Bayesianism Pays...Sort Of

According to Bayes's Theorem, there was no rational reason to watch the Fourth Game of the World Series.
P(h|e) = [P(e|h)P(h)] / [P(e|h)P(h) + P(e|not-h)P(not-h)]
Forgetting about alleged differences between the National League and the American League, assume a prior probability that the Sox will win the series as basically a 50/50 proposition, and let each successive win or loss increase or decrease our confidence by an equal amount:
P(h) = .5
e = +/- .125
Thus, after Game 1:
P(h|e) = (.5)(.625) / (.5)(.625) + (.5)(.375) = .625
After Game 2:
P(h|e) = (.5)(.75) / (.5)(.75) + (.5)(.25) = .75
And after Game 3:
P(h|e) = (.5)(.875) / (.5)(.875) + (.5)(.125) = .875
Now, I submit that when there is virtually a 90% chance of rain in the forecast, one takes one's umbrella to work. Similarly, when you are up by three games, you ought to win the Series.

Unless you are the Cleveland Indians.

Subjective probabilities may give us some insights into the gambling behaviors of "idealized persons", but clearly they are not the same thing as objective probabilities (or propensities). In the present case, the Sox did go on to win Game 4, thus brining the probability of their victory to 1, but of course the model only tells us what we may reasonably expect, not what is going to happen, and they could have gone on to lose it all, just as the Indians lost the ALCS in spite of having a .875 chance of winning it all.

Some probabilistic models, however, do not model our own epistemic limitations, but actually reflect the deep structure of reality. As John Bell's work in the mid-1960s showed, there is no possible "hidden variables" account of Quantum Mechanics that can account for the "mysterious correlations" unless we either assume that QM is incomplete or abandon locality. Neither option seems particularly attractive, so what to make of Bell's work?

There is an alternative, of course: abandon determinism. Why do we assume it in the first place? Because, obviously, without it the idea of science as a predictor rather than a mere describer becomes rather more complex than some folks would like it to be. But abandoning determinism globally does not mean that we must abandon it at the methodological level. Local determinism is certainly true within certain frames of reference, whatever we may want to say about the behaviors of subatomic particles. But must we assume that it is true in every frame of reference above the level of subatomic particles? Evolutionary theory is replete with formulae expressing probabilistic outcomes (fitness is the classic example)--are they merely epistemic probabilities, or do they reflect something fundamentally true about the order of nature? Can fitness, for example, be thought of as an objective probability?

Frankly I don't see why not. The determinist can give no non-question-begging argument to prove that there exist "hidden variables" behind these probabilities. I don't think that the indeterminist can give anything like Bell's proof that "hidden variables" explanations are not possible, but I do think that we have rational warrant for supposing that at least some probabilities in Evolutionary Theory reflect more than just our own epistemic limitations. If so, there is reason to think that at least one variety of scientific realism may be false, and that is a good thing.


John Farrell said...

Proving yet again how baseball is, deep down, like life. (although I know that wasn't your main point)

Scott Carson said...

Well, here in Indians country, anyway, life can be full of surprises--and disappointments. Though I have to confess that I've been a secret Boston fan since before I moved here. Same with Patriots football. I used to pull for the Browns, but that got to be far too difficult after 1987.

Tom said...

According to Bayes's Theorem, there was no rational reason to watch the Fourth Game of the World Series.

You know, there are people who actually believe such things. I mean, who really think statistics apply to singulars, who think a statement like, "I have a 68% chance of living past age 75," is both meaningful and true.

Scott Carson said...

Shoot! I was counting on there being a non-zero chance of living into my 100s. Now you're telling me that, even if there is, I still might not make it! Way to ruin my day.