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Tuesday, January 16, 2007

Probability

The more I think about it--and I'm a whopping thirty years old, ya know--I don't think it makes sense to define probability. I think you either know what "1 in four chance" means, or you don't.

(1) It doesn't mean, "If you did this 1 billion times, then..." It's statistically possible that you could get 1 billion heads in a row with a fair coin. So you can't talk like that to define 1/2 probability.

(2) I used to think it could mean, "You would be willing to give x odds in a wager on the subject." But no, that doesn't work either, because of both incentive problems and "diversification" issues (for lack of a better term). E.g. if I say there's a certain probability I'll miss my connecting flight (we were in the hotel between legs of the trip home when I thought of this), that doesn't translate into the odds I'd need to wager against myself.

Obviously, I would always be willing to bet that I'd miss my connecting flight, because I can just walk very slowly. On the other hand, I might not want to bet the other way, because who wants to get a big payoff if he makes his connection, but lose money if he misses it??

26 comments:

  1. Anonymous4:18 PM

    Would you agree that this is a source of difficulty for "hardline" Austrians, who disdain all use of mathematical models to better understand economic problems?

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  2. I'm not sure what you mean. I actually was expecting a Hoppeian to tell me that Ludwig von Mises had satisfactorily defined probability in Human Action.

    ReplyDelete
  3. Anonymous2:22 AM

    I'm not sure what your thoughts are on Caplan's paper about Austrian economics is, but it seems to me that certain Austrians, Hoppeans being a great example, are unable, to grok certain nuances of economics, such as probability (psychological or otherwise). For example, if someone said that they felt about 80% sure that they would take a certain course of action, from what I understand a rigid praxeologist would not find such a statement meaningful, since it isn't necessarily expressed through action.

    Maybe I read too much into your post, but I do think that probability is an area of blindness for economists that disdain *any* benefit in using mathematical models.

    And as I reread my post I also realize that there are many senses of probability, very different from one another, but they are not necessarily right or wrong, but context-dependent, and all seem impossible to define without the aid of metaphor and example.

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  4. Anonymous2:27 AM

    I estimate that the probability that the "is" in the first sentence of the proceeding post will continue to bother me as long as the thread continues is high.

    ReplyDelete
  5. Oh okay. That's what I thought you were asking, but I wasn't sure, since it seemed weird that in response to a post where I talked about the difficulty of defining probability, you were wondering if I agreed that the Austrians should use probability.

    How's this for an evasive answer: I think the Austrian purist is right that the attempt to model uncertainty in mainstream economics makes a bunch of unrealistic assumptions. On the other hand, I disagree with the Mises/Hoppe dichotomy between case and class probability.

    For example, I don't think it's nonsense to talk about the probability of Hillary Clinton being elected president. If I recall correctly, Mises and Hoppe would say that would be a unique event and so we can't talk about its probability of occurrence.

    ReplyDelete
  6. Probability is, I think, an epistemological issue. Take a fair coin toss: If you were in a position to know all of the relevant force vectors at the time of the toss, then you'd be certain how it would land -- the odds wouldn't be 1/2 at all, but 1. Mises was working from his brother's famous frequency theory of probability (btw, I've met several people who had heard of Richard von Mises but never Ludwig), which was a good try, but is generally seen to have come up short.

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  7. Anonymous2:27 PM

    The two philosophical schools are the frequentist (if you do it "enough" times....) and the Bayesian (probability is a matter of belief, refined by experience).

    Hard subject. I tend towards the Bayesian.

    ReplyDelete
  8. Anonymous11:20 AM

    Doesn't it depend on context? I think, Gene, that you are more or less right about it being epistemological, but this is almost trivial. There clearly are different kinds of probability, I don't see that frequentist's are wrong, necessarily, but that Bayesian probability can be a useful framework.

    Bob, I think you are more or less right that modeling uncertainty is an, uh, uncertain enterprise, but it seems to me that this alone doesn't make it invalid. You can use metaphors, even if they are not strictly "true", to model natural phenomena. For example, isn't it possible that a business crash could occur based on irrational probability judgments?

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