Comments on: Random Finds in Heterodox Economics, #2

By: bill carone

bill carone — Sun, 22 Feb 2004 20:48:12 +0000

“Pascal’s wager, if I understand it correctly, poses a question where there’s a binary choice without a probability measure.” I’ve checked with a few sources, and it seems that any assignment of probabilities (p) and (1-p) to possibilities (A) and (not-A) is a probability measure. Let S be a set containing A and not-A. Look at this collection containing these subsets of S: – empty – A – not-A – A, not-A (i.e. the set S) These form a sigma-algebra, since – S is in it, – If X is in it, so is its complement, and – The algebra is closed under any countable number of set operations. The measure that assigns the following measures to the subsets is a probability measure. – p(empty) = 0 – p(A) = p – p(not-A) = 1-p – p(S) = 1 Do you see something wrong with the above?

By: bill carone

bill carone — Thu, 19 Feb 2004 17:12:37 +0000

“Pascal’s wager, if I understand it correctly, poses a question where there’s a binary choice without a probability measure.” Do you have a source on that? (I can’t find a source on Google) It doesn’t match my understanding of a probability measure. I’m no expert, though. This is the wager where either an afterlife exists or not, and you choose to be religious or not? I thought the problem was an infinite utility (or multiple infinite utilities), not about probabilities. “In general, I think we’re talking at cross purposes.” Sorry, I am clearly misunderstanding; first I thought it was about chaos theory, then game theory, and now it is probability theory. “For example, game 3, which is the same as game 2 but it’s my brother, about whom you know as little as you know about me, doing the rolling. What I think Keynes and I would say would be that there is no principled way in which you can say either that gamble 2 is preferred to gamble 3, or that you are indifferent between them.” I don’t see why either of you would think that. Here is the principle: If you have the same information about two possibilities, then you should assign the same probabilities to them. In other words, if every piece of information you have points equally to A as to B, then p(A)=p(B). This is what I mean by the “principle of insufficient reason”; why would you not use it in this case? My reasoning goes as follows: – my probability of winning game 2 equals my probability of winning game 3 (whatever it is). Why? You have specifically told me that I have no information that distinguishes your action and your brother’s action. – Same probabilities, same outcomes implies indifference. Intuitively, if I had to play one, I’d be indifferent between the two; I have no information one way or the other about which is more likely to win. This also is the intuitively correct result, no? How could it be otherwise? “What we’d question is whether probability in this sense is something that should be put on the same logical footing as frequentist probability” These probabilities work exactly the same way as frequentist probabilities, as shown by Cox; see Jaynes. Briefly, if you represent your information as real numbers, and you want to reason consistently, then he shows that those real numbers must obey two rules: p(A|C) + p(not-A|C) = 1 p(A and B|C) = p(A|C)p(B|AC) These are the same rules that frequentists use for finite, discrete sets of possibilities. So probabilities representing your information combine in the same way as frequencies do.

By: dsquared

dsquared — Thu, 19 Feb 2004 15:20:57 +0000

Can you give me an example of a finite set without a probability measure? Pascal’s wager, if I understand it correctly, poses a question where there’s a binary choice without a probability measure. In general, I think we’re talking at cross purposes. Neither myself nor Keynes would disagree that you can assign a probability based on your perception of my honesty. What we’d question is whether probability in this sense is something that should be put on the same logical footing as frequentist probability; the probability that a dice turns up the right number. For example, game 3, which is the same as game 2 but it’s my brother, about whom you know as little as you know about me, doing the rolling. What I think Keynes and I would say would be that there is no principled way in which you can say either that gamble 2 is preferred to gamble 3, or that you are indifferent between them. This is Keynesian probability; it’s a theory of Bayesian probability-as-degree-of-belief, but it doesn’t require a complete ordering.

By: bill carone

bill carone — Wed, 18 Feb 2004 21:27:35 +0000

Dsquared, First, a sincere thanks for the response. “For the first, the PIR is going to give you wrong answers when you’re trying to make guesses about a set of outcomes which doesn’t have a probability measure.” Can you give me an example of a finite set without a probability measure? In the Pruss paper you linked, he starts with infinite sets: “Suppose I know with complete certainty that X could have any value in the interval [0,1], and there is no reason to favor any value over any other.” “Suppose, however, that (a) I believe the universe is infinite in extent,” When you start with a finite set, then take a well-behaved limit to an infinite set, then probability calculations always work; they give the correct (usually intuitive) answer when such a limit exists, and refuses to answer when it doesn’t. For example, in Pruss: “For instance, a quick calculation shows that the likelihood that the blast will occur within a thousand miles of Earth is less than 10^{-44 of the likelihood that the blast will occur somewhere in the Andromeda Galaxy. Any likelihood that is that much smaller than some other likelihood is negligible.”

“Admittedly, all the reasoning here cannot be reconstructed within a standard Bayesian epistemology based on the classical probability calculus.”

It can; all you do is start with a finite universe and take a limit to an infinite universe; in the limit the first probability is less than 10}-44 times the second probability. You get the same intuitive answer as Pruss, using standard probability theory. The limit method always works, and it satisfies our intuition about “infinite” cases when the limit converges. When the limit isn’t well-behaved, the mathematics is telling you “You can’t use this shortcut on this problem; you can’t just say N is very, very large, you actually have to figure out how big you think N is.” The St. Petersburg case is an example; the limit diverges (increases without bound). This means you can’t conclude anything about the answer; you can’t just say “Imagine we could play forever,” you need to be more specific. Once you put a limit on the amount of money or the number of plays, intuitive answers pop out. You can solve most (all?) “infinite” paradoxes this way: Do all the analysis with finite quantities, then see what happens when those quantities increase indefinitely. This is the method Gauss used: “I protest against the use of infinite magnitude as something accomplished, which is never permissible in mathematics. Infinity is merely a figure of speech, the true meaning being a limit.” 2. So it isn’t chaos theory I should be looking into, it is just game theory, right? Game theory often upsets me for similar reasons to the stuff above: it assumes an infinity already exists (an infinity of anticipations). But I understand it much more than I do chaos theory. “probabilistic reasoning in the sense of assuming well-defined and objective probability measures” I don’t know what you mean by “objective” here. See below. “2. I throw a dice behind a screen, which you’re never allowed to see. About 1 time in 6, so far in plays of this game to date, I say “Hey, you rolled a 6” and pay you 20x your stake, and about 5 times in 6 so far I have said “Sorry, better luck next time” and kept your stake.” So I don’t know if you are honest or not here? The “five times out of six” is relevant, but not a complete description of what I know. Probability is about all of my information. My judgment of your honesty would enter into the probability calculation. For example, if you were my friend and the money would go to Bill Gates if I lose, I would assign a different probability than if you were a stranger and the money went to you if I lose. It would also be different if you didn’t know the bet was for $1 million this time (or if all the bets in the past were for $1 million). I’m missing the point you are making here: – I don’t see a problem with my assigning a probability representing my information, then using standard decision theory. – I also am missing any game-theoretic issues (“he thinks that I think that he thinks …”) as in Rosser’s paper.

By: dsquared

dsquared — Wed, 18 Feb 2004 13:52:53 +0000

Bill: (sorry for delay). For the first, the PIR is going to give you wrong answers when you’re trying to make guesses about a set of outcomes which doesn’t have a probability measure. ( For example, you can’t make sense of the Argument From Design or Pascal’s Wager based on probability theory for this reason). 2: It’s implicit in the Rosser papers that probabilistic reasoning in the sense of assuming well-defined and objective probability measures is often going to come to grief in situations where there is a lot of game-theoretic structure. For example, think about the difference between the following two bets: 1. I throw a dice in plain view and pay you 10x your stake if it lands ‘6’. 2. I throw a dice behind a screen, which you’re never allowed to see. About 1 time in 6, so far in plays of this game to date, I say “Hey, you rolled a 6” and pay you 20x your stake, and about 5 times in 6 so far I have said “Sorry, better luck next time” and kept your stake. You’ve just inherited $1,000,000 with the condition that you have to risk it all on one spin of a game of chance. Which of the two options above would you risk it on? The risk here is your assessment of whether I’d rip you off or not. I don’t see how there’s any rational principle at all on which you could make the decision between these gambles, but that doesn’t mean they’re equally attractive.

By: bill carone

bill carone — Wed, 11 Feb 2004 17:24:55 +0000

“When you assign a uniform distribution over outcomes because you have no information (as you explicitly suggest above), that’s a version of the principle of insufficient reason.” Well, there are lots of rationales for it: transformation groups and maximum entropy are other ones that lead to the same sorts of conclusions. See Jaynes ; examples of transformation groups leading to uniform distributions are in the section beginning on page 17. “Most of the time it will work. Sometimes (particularly when dealing with complex systems), it could be very bad indeed.” I would be eternally in your debt if you would answer two questions about this (ever since your post, they are driving me crazy): 1) Can you give an example where the principle of insufficient reason doesn’t work when assigning probabilities to a finite set of possibilities? I’m pretty sure none exist, but I’m willing to learn. This usually clinches it for me; probability theory (and I believe, with far less confidence, mathematics) are about finite sets and well-behaved limits of finite sets. 2) Can you give an example of one of the complex systems when it is “very bad indeed” to assign equal probabilities if you have no information. I thought you said I would find an example in one of the Rosser papers, but I couldn’t; if you could point me to one in there, or another you know about, I would really appreciate it.

By: bill carone

bill carone — Wed, 11 Feb 2004 17:01:11 +0000

Dave, thanks for responding. “It’d be good to be clear on simple things, like time value of money, before worrying about things like labor theory of value, or cauchy distributions” That’s fair. Let’s see now. “In the example given, as far as I can tell, the capitalist is simply breaking even; we’ll chain the other direction this time:” Yes, if the capitalist has a 5% time preference as well (like the workers), then he breaks even. If he has a smaller time preference, he makes a profit (is it commone for capitalists to have smaller or larger discout rates than workers? Workers need the money, but capitalists have more investment opportunities?) However, that was not my point. My point was that he made any monetary return at all on his original investment at all without putting in any labor. I was confused by the fact that the capitalist could make a 5% return on investment without putting in any socially useful labor. I thought that adherents of the labor theory of value (LTV) would object to that. I then showed that it was equivalent to another distribution only to the workers that a LTV adherent would, I thought, accept. D^2 then, I believe, put me right when he told me that the LTV doesn’t take a position on distribution, but I wanted to hear what other people had to say as well. So, would you object to the capitalist making 5% on his investment without putting any labor into it? “it doesn’t take a cauchy distribution for means to not be very useful” True; I only meant that Cauchy distributions frequently give strange results when dealing with means and variances, rather than some other summary measures. It is usually the catch-all counterexample to anything involving means and variances (e.g. law of large numbers, minimum variance unbiased estimators, etc.). “theoretically, projects are supposed to behave like they have expected completion dates, and naively one might expect that the longer one has gone without observing completion, the more likely completion will be in the near future” I’m not sure this is true, even theoretically. For example, if a project will be completed in an exponential amount of time, then the expected time (and the distribution of the time) until completion never changes (Dilbert would be proud :-) This is a standard fallacy when means are involved. “the longer many projects have gone on without success, the farther in the future their expected success will be” Again, I don’t think this is poses any theoretical problem; I teach this in first-quarter probability. Many lifetime distributions have similar qualities. For example, when I buy a computer game, the expected length of time until it breaks down starts out at X, say. Once I actually get it installed and start playing, the expected time until it breaks down increases dramatically; it “wears-in.” Isn’t that what you experience? Another example; my wife will have a child soon; when it is born. I will assign X as its expected life. A year later, the expected life will have increased, no?

By: bill carone

bill carone — Wed, 11 Feb 2004 17:01:10 +0000

By: dsquared

dsquared — Wed, 11 Feb 2004 16:03:02 +0000

Bill: When you assign a uniform distribution over outcomes because you have no information (as you explicitly suggest above), that’s a version of the principle of insufficient reason. Most of the time it will work. Sometimes (particularly when dealing with complex systems), it could be very bad indeed.

By: Dave

Dave — Wed, 11 Feb 2004 04:51:53 +0000

It’d be good to be clear on simple things, like time value of money, before worrying about things like labor theory of value, or cauchy distributions*. In the example given, as far as I can tell, the capitalist is simply breaking even; we’ll chain the other direction this time: Y1 -100 = Y1 -100 Y2 -100 = Y1 -95 Y3 +215 = Y1 +195 * a) it doesn’t take a cauchy distribution for means to not be very useful. The mean is simply the first coefficient in an approximation, and trying to treat highly skewed distributions with only the mean is somewhat like approximating addresses with only latitude. b) the cauchy distribution is well known to those of us who have ever worked someplace where “Dilbert” is popular: theoretically, projects are supposed to behave like they have expected completion dates, and naively one might expect that the longer one has gone without observing completion, the more likely completion will be in the near future; practically, as Schroeder points out, the longer many projects have gone on without success, the farther in the future their expected success will be. CS people know about Real Soon Now, and folk wisdom knows about Mañana.

By: bill carone

bill carone — Tue, 10 Feb 2004 19:45:35 +0000

Comment on Rosser “Epistemology …,” especially about knowledge of chaotic systems. “One simply cannot guarantee exact prediction, or even very close prediction with any certainty, as long as one is expending a finite effort to obtain information regarding the system, its internal relations, its initial conditions, its parameter values, and so forth.” Is he confusing complexity with uncertainty here? For example, if I assign a distribution on initial conditions, parameter values, and so forth, then for each possibility, I can compute exactly what will happen in the system, even though it is chaotic, no? Then, I use standard Bayesian calculations to figure out a distribution on what will happen. The end result may be (in fact, in a chaotic system, most likely will be) that I know very little about what will happen, but there is no problem assigning probabilities that describe that ignorance precisely, is there? In other words, I don’t see an argument for any epistemological problem with assigning probabilities over the results of chaotic systems. The result I see is that, with chaotic systems, the calculated probabilities will indicate ignorance. Can you point out a specific example that I can look at to understand what you mean when you said “it is not always obviously legitimate to use the Principle of Insufficient Reason in cases of ignorance.”

By: bill carone

bill carone — Tue, 10 Feb 2004 16:40:54 +0000

Comments on “All I have to say has already crossed your mind,” especially Argument II I haven’t been able to get the Morgenstern, Clayton, or Diaconis papers, so perhaps they explain some of the following. “The essence of the result we derive below has been expressed by Clayton 1986 (p. 38). Imagine “A and B are witness to some coin tossing. Bayesian A is firmly committed to the belief that all coins are fair, and so uses d1/2 for q, the probability of heads. B is firmly committed to the belief that coins are never fair, and uses a uniform prior on [0,1/3]U[2/3,1]. Both A and B will use Bayes theorem to coherently update their priors as they see data, but they will never agree, nor should they” Probabilities should represent information, not mere belief. Before I start calculating, I would want to know what information A has that committed A to A’s belief, and the same for B. Note that all it takes to destroy the result is the minimal claim “I might be wrong,” putting a small density on all points between 0 and 1. It doesn’t seem that this is the essence of Rosser’s argument II. Let me try to summarize it. You are a single decision-maker, trying to decide what to do. The problem is, there is another person who is trying to outthink you; your opponent will anticipate your action, or anticipate your anticipation of his anticipation of your action, etc. Let n be the number of your opponent’s anticipations (i.e. the two examples in the previous sentence are n=1 and n=2). Now, if you knew n for sure, then you could choose the optimal decision (which in this case is a particular mixed strategy). However, as n gets large, the optimal decision might not converge, but instead oscillate between two choices. Rosser then (erroneously, I think) concludes that there is no solution to the problem, so all such decisions must be made, not by careful rational analysis, but by animal spirits. Two problems I see with this argument. First, the solution might converge; in his example it doesn’t, but in others it does. Then, some decisions could be made using rational analysis. The example given involved the words “Cauchy” and “posterior mean” which set off alarm bells in my head (Cauchy distributions are famous for not having a mean, so often it can cause hidden problems when you try to use means as summary measures). It also uses continuous, infinite-ranged probability distributions, which often cause problems when people don’t remember that they are just limits of discrete, finite-ranged distributions (this is quite common in the probability literature, and causes all sorts of nonsense to be published; people forget that infinity isn’t a real number, either for engineers or mathematicians. I’m not sure if Diaconis or Rosser are particularly susceptible to this sort of thing). Second, if the mathematical limit does not converge, you can’t make any other conclusion other that “my model (or the problem) is ill-posed.” The model above says “We don’t know much about n, other than it could get large. Let’s not think about it too much, and just see what happens when n gets really large. Maybe it will give us a sensible answer.” When the limit doesn’t converge, you should then refine your model; the divergence tells you, “Look, this particular “n gets large” shortcut, although it works a lot of the time, isn’t open to you in this case. Go back and do some more work.” In this case, the “more work” that is needed is a distribution on n. After all, no one actually anticipates infinitely often. So figure out what you know about your opponent’s anticipation strategy. My friends often win games this way (“Bill will most likely think either three or four moves ahead”). Perhaps we could put a uniform distribtuion on all possible n from 1 to N, then see what happens when N goes to infinity. In the example in the paper, where for all even n the posterior mean was y and for all odd n the posterior mean was ~~y, such a limit would give a posterior mean of zero (which, it seems, is the solution Rosser wants it to give). So, instead of non~~convergence = no answer, non-convergence = bad model that needs to be fixed. Here, instead of assuming n is large, actually model what you know about it. Now there may be a little problem with my idea. This strategy I’ve suggested might just be a subset of the strategies already considered. In other words, adding this “put a distribution on n” step in the argument might lead to a whole other anticipation game, as he anticipate what n-distribution I am usign. I don’t think this is the case; here, the n-distribution describes my information, whereas before, the mixed distribution described my actions. However, the information about n depends on what he thinks about my distribution on n, so there still may be a problem.

By: dsquared

dsquared — Sun, 08 Feb 2004 23:01:17 +0000

That does not gibe with my experience in the financial sector. Most arbitrageurs don’t have the deep pockets of a George Soros or an LCTM. Yeh, but come on … you don’t need anything like the size of bat those guys have to be able to move prices.

By: dsquared

dsquared — Sun, 08 Feb 2004 22:23:34 +0000

The mathematical sense. It’s now settled ground that the approach taken by the Cambridge (US) side – that one could measure the capital stock taken as an aggregate by using the dollar prices of capital equipment – is not satisfactory given that it means that anything you then go on to say about the rate of profit is a circularity (because the price of capital equipment depends on the rate of profit). There is some room for dispute about how important this is (see above) but the central point is purely mathematical.

By: Anarch

Anarch — Sun, 08 Feb 2004 21:33:18 +0000

The great thing about the capital debate is that one side is provably right, and the other side is provably wrong… Provable in what sense?