Friday, December 06, 2002


LI learned our probability theory from the Dover Press edition of Richard Von Mises book on same. At the time, we did not realize that Von Mises was presenting a much controverted thesis on probability -- that he represented the extreme point of the extensional school. von Mises was the brother of the conservative economist -- although, according to his biography, he was definitely not hedged about by his brother's libertarian ideology. He gave up an honor given to him by East Germany with the admission that he would have taken it if the times -- the year was 1952 -- didn't make any truck with the communists automatically suspicious.

James Rizzo, in this essay on expected utility, gives a good overview of the difference between the view that that probability refers to the frequency of the observation of an event's occurence in a series of observations, and the 'subjectivist' view, which makes a softer case for the meaning of likelihood.

"Probability is neither a simple nor innocent concept, and there have been profound disagreements, especially during the 20th century, over basic matters of definition. Although I have relatively few original things to say about the terms of these debates, my discussion cannot proceed without minimally outlining them -- for probability is where, I am arguing, decision theory stows its metaphysical baggage. I am not sure how obvious my basic point that probability is a metaphysics might seem. On the one hand, it is clear that even our everyday concept of "probability" depends on fairly specific claims about the nature of the universe (the cosmos) and its knowability. And Ian Hacking?s (1975, 1990) efforts to relate the emergence of probability to various modernist projects, like the building of the nation-state, are well-known. On the other hand, critical social theory and Marxism have paid far less attention to probability than it deserves -- if we take its metaphysics as seriously as I propose we do.

A first step in this direction would be to account for the opposition between the frequentist (objectivist) and personalist (subjectivist) definitions of probability.

In large part, frequentism represents an extension of the classical theories of Laplace and Pascal, in which probability was treated as a ratio of favorable to equally possible cases --the paradigmatic events here being series of coin tosses, dice rolls, and other recreations of the French aristocracy. Modern objectivism treats probability as the limiting value of the relative frequency with which certain events, or properties, recur within a sequence of observations. Most frequency theories (such as the one advanced by Richard von Mises) do not require this sequence of observations to be finite, i.e., it can stand in for limiting relative frequency that would be manifested by the unlimited repetition of the event. Peirce, whose theory of probability is in many respects frequentist, is quite clear on this point:

"Probability never properly refers immediately to a single event, but exclusively to the happening of a given kind of event on any occasion of a given kind. [I]tis plain that, if probability be the ratio of the occurrences of the specific event tothe occurrences of the generic occasion, it is the ratio that there would be in thelong run, and has nothing to do with any supposed cessation of the occasions.This long run can be nothing but an endlessly long run..."

Hans Reichenbach, who was also a logical positivist, was dissatisfied with a position that seemed to rule out saying things about singulars - like giving the probability of landing on Mars at a certain date. A professor Uchii has a nice site sorting through these issues. Reichenbach's compromise basically gives us a concept of possible worlds -- thus embedding a theory of probability in what will later, under Kripke, become a theory of description: i

"According to Reichenbach, the probability concept is extended by giving probability a "fictitious" meaning in reference to single events. We find the probability associated with an infinite sequences and transfer that value to a given single member of it. ... This procedure, which seems natural in the case of the coin toss, does involve basic difficulties. The whole trouble is that a given single event belongs to many sequences, and the probabilities associated with the different sequences may differ considerably. The problem is to decide from which sequence to take the probability that is to be attached "fictitiously" to the single event."

So: the point, here, is that when we are making probability claims, we have to get our theory of probability straight. And a refined version of extensional probability, one that can encompass a single event, still needs to construct a a reference class and an attribute class. The attribute class is some definite description, and the reference class is the particular, defining order of events or properties under which to classify our observations. Got that?

So what, pray tell, is William Saletan doing with his Saddameter in Slate?

The premise is the jokey one that invading Iraq is much like Wheel of Fortune -- an idea reinforced by the visual. This is, of course, in accordence with the idiosyncratic Saletan touch, tasteles and tacky, a subdeb Harvard Lampoon conceit. But it is also a completely odd exercise. Every day Saletan gives us the "odds" on invading Iraq. Well, what does this mean?

The problem is that the relationship to a reference class, here, begs the question: what is the reference class? Let's try to think this one through.

100% must refer to the certainty of invasion. But, if this is so, what does 0% refer to?

On the one hand, LI could make the case that, unconsciously, Saletan has constructed a reference class that includes all the non-USA nations. We can then assign hostility quotients to them -- Canada, for instance, would get so much, and Syria would get so much, and so on. Thus, the probability of invading Iraq would refer to the class of invadable nations.

But we doubt this is Saletan's point. Although he believes that odds talk is self-explanatory, LI thinks that Saletan's assumption is much more revealing than his exercize. The relevant reference class, in LI's opinion, is the punditocracy sense of the certainty of an Iraq invasion. The odds, in other words, refer to another level of odds. And that refers to the penchant, among the punditry, for belligerence or pacifism. So 100% would be, say, the Weekly Standard editorial board, and 0 would be,, say, Hans Blix p.r. man. With the in betweens probably being those who are pacifistic but think the US will invade Iraq, those who are belligerent but think Bush will chicken out, and so on.

You'll notice the large divergence between the reference classes. They don't, actually, share any members. Well, this doesn't surprise us. Saletan, for all his snobbery about the great unwashed that live outside his zip code, has never shown himself to be a very bright bulb himself. That the odds thing continues to take up space on the Slate site is a little amazing to me, however, since Slate prides itself on running nit-picky pop sci features that knock down buncomb in other forums.

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