I choose option B (assuming the coin flips are fair. Preferably we'll use a source of quantum noise ). But if you make the numbers $1,000,000 and $50,000, I'll probably choose A. Between those values I'd have to think hard.

The EV of B is infinite (I think it's actually a cool example of a "pascal's mugging"), but the expected utility is not-- as stated, people run the $ through a (probably bounded) utility function, and while $2 million is better than $1 million, it's less than 2x better.

Anyway, cool discussion. One thing does not make sense: How could Keynes possibly be Bayesian and think that there is no possible estimation of lifetime earnings at the same time?

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I suspect that you're right in that option B is the equivalent to thousands of tickets, but it's still a 3% chance to get more than $10. I would choose option B over ANY lottery.



Yeah, I can understand that. However, I did some quick math and 93.75% of the time you'll want to take the first option since you'll end up with less than $10, and 6.25% of the time take the second option since you'll end up with more than $10.

I fully recognize that from a pure mathematical view option B is superior. The person could throw 500,000,000 heads in a row. Likely, no, but it's still possible. In a mathematical setting, I'd choose option B everytime. In practice, I'd choose option A about 94% of the time.

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Bill is being controversial Don't blame poor Keynes. He definitely thought that people estimate their lifetime earnings. He just thought that those estimates (indeed, most economically relevant estimates) were uncertain, and thus prone to sudden sharp changes that were not, strictly speaking, warranted or unwarranted from a logical point of view.

This is a stab at an explanation, but I don't have time to finesse it or double check what I'm saying so I hope some of you will correct/improve it.

Keynes inaugurated the distinction between risk and uncertainty. The distinction between risk and uncertainty implies that in some cases, even with all the data that it's rational us for us to gather, we can still be wrong about the true probability of the event. However (ripostes Ramsey or someone), the "true" probabilities of events in a (fairly) deterministic universe is always 0 or 1, and the only way that it makes sense to talk about probability is as whatever number we've decided on after we've gathered all the data we care to collect. Thus, there's never any uncertainty. As I understand it, that's why Keynes is grouped as a Bayesian and his anti-uncertainty allies are grouped as frequentists. However, it's confusing because the anti-Keynesians were the ones who pioneered the idea that you could determine what probability someone really assigned to something by getting them to bet on it, which is currently a fad that (on the internet, at least) groups with calling oneself "Bayesian".

Alright, I still don't see why one's estimates can't take take all that into account. It's the whole point of doing probabilistic math, to quantify your ignorance...

Thanks for the explanation.

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Well, you'd want to read the Treatise on Probability to see why Keynes wanted to distinguish between risk and uncertainty for mathematical and logical reasons, and the General Theory to see why he found his earlier views on probability useful for economic analysis (Chapter 12 is most on point, iirc).

You are told that an urn has some unspecified combination of white and black balls. What is the probability that the next ball you draw is white?

The natural response is "I got no clue" but it's actually really hard to represent that

Or, even better, try to represent your probability estimate that Goldbach's conjecture is true. As a mathematical claim, it had better be 0 or 1, but that certainly doesn't represent your epistemic situation. This goes by the name of "the problem of logical omniscience."

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First, being a Bayesian simply means that you assign probabilities to propositions or hypotheses. Keynes regarded probabilities as logical relations between propositions. He did not think that all probabilities are numerical. He also refers to "vague" knowledge, which I think is what we now would call "fuzzy", and did not think that it was "susceptible of strict logical treatment" (A Treatise on Probability, p. 17). Bayesians these days, I suppose, tend to think of probabilities as numbers and rely upon Cox's Theorem.

I think, then, that Keynes would certainly allow a vague estimate of lifetime income (as vague knowledge), which I think that my example of the woman who plays the lottery illustrates. As for how to de-fuzzify that estimate, well, de-fuzzification is a question, isn't it? And the result of de-fuzzification would not be a mathematical expectation, would it?

And even if we were talking about probabilistic knowledge, I think that Keynes would reject the idea that we could get a reasonably precise estimate. (He discusses such matters in pp. 24ff, and I cannot claim to do him justice in a few words. ) By a reasonably precise estimate I mean one which is bounded closely by two numbers. (Keynes allowed that non-nummerical probabilities could be bounded by numbers. That is obvious when you note that all probabilities lie between 0 and 1. ) He talks about a lawsuit in which a racehorse was sold, and therefore a contract to have it service a mare one summer was breached. The judge awarded only nominal damages, because there were too many contingencies involved to award more. OC, when we are talking about lifetime income, the contingencies are far greater.

But what about statistics? Keynes mentions the case of Gibbon estimating his life expectancy from actuarial tables. Keynes remarks, "But if a doctor had been called to his assistance the nice precision of these calculations would have become useless" (p. 29).

In summary, "Whether or not such a thing is theoretically conceivable, no exercise of the practical judgment is possible by which a numerical value can actually be given to the probability of every argument" (p. 27).

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I do not know much economics, but this is from Wikipedia ( http://en.wikipedia.org/wiki/Consumption_function ):

"The Keynesian consumption function is also known as the absolute income hypothesis, as it only bases consumption on current income and ignores potential future income (or lack of)."

A couple of points. First, changes in disposable income track changes in consumption expenditure quite well. Second, you can see why Keynes might have ignored potential future income: too many contingencies.

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I always try to find something by Cosma Shalizi when I'm reading or thinking about anything related to statistics, and this is the result today: http://bactra.org/weblog/612.html.

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Random question: Can the balls problem actually be solved theoretically? In practice, I'd just solve it experimentally.

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I'm with you (I think) in believing we can usefully quantify a lot of things about the world, and estimates can take a lot into account, but our probabilistic math can only be put into the service of our properly stated propositions, whose relationship to the world is nightmarishly complicated. The "sudden sharp changes" above reminded me of the breen/glue thought experiment I read about somewhere. A "breen" object is one that appears green except on alternative March Thursdays in a Leap year when the Russell 5000 is up more than 20% in the previous quarter, at which point it appears blue. A "glue" object is one that appears blue most of the time, but looks green during those same Thursdays. A quick survey of trees here in LA indicates to a very high confidence that the statement "leaves are breen" is true, and yet it is painfully false. It seems silly, but those workers estimating their future income and assuming, probability 1, that they "had" a job (a "green" job?) would be shocked to find that the next leap year, the Russell having done well, Bain bought their company and they were fired. (Now they're "blue," perhaps?)

I would take any amount that compensated me for the worry about losing, and the trouble involved in making the bet- say $105.00 and up. I think five bucks is about right for me.

This is only because losing $100.00 wouldn't affect me very much. I couldn't generally tell you my checking account balance to within $100.00 (I'm not rich, I'm just careless.) If you made the bet $1000.00 I would need much better odds to be willing to take it, and if you made it $10,000.00 I would require very favorable odds, and a patient payment plan, assuming you had some way of enforcing payment. If you couldn't enforce payment that would change things .)

Rational, in an economist's sense, is not exactly the same as rational in real life. This is not necessarily a problem with economics- it is simply down to the fact that there are factors individuals have to consider that are better abstracted away when dealing with people in the aggregate.

She may be right about new Bayesians, those who have come up after the Bayesian revival of the last half of the 20th century, but that remark does not apply to Keynes.

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How do you solve it experimentally? Will you draw one ball and if it turns out to be black, will you say probability is 1?

That makes me wonder about the terminology: unfortunately there's a lot of different ways one can count as a Bayesian, and I can't pretend to know them all.

Side note: Cosma is a he.

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A he. Oops! I was thinking Cosmo-Cosma.

I think that there is a clear definition. A Bayesian is someone who believes that there are probabilities of propositions.

One thing that follows from such a belief is a belief in confirmatory evidence. I. e., evidence can increase one's degree of belief in a proposition. Confirmatory evidence is very weak. Most people believe in confirmatory evidence, and it is hard to see how our judicial system would function without that belief. Nowadays you hear professional witnesses use the phrase, "is consistent with", on the stand, and TV police say that they want to "eliminate {someone} as a suspect". Both phrases, OC, come from Popperian science, which seeks to disconfirm hypotheses rather than to confirm them. But really, police and prosecutors are looking for confirmatory evidence.

One attraction of Bayesianism is that it is logical. Frequentist statistics is rife with ad hockery. But the application of Bayesianism has always been problematic.

I think that the idea of a prior probability distribution of the form, "The probability of P is x", where P is a logical proposition and x is a real number between 0 and 1 is due to Laplace. That has always been problematic. If it were not, we would never have had the Popperian-Fisherian revolution in the first place. We would all be be happy Bayesians making inductions with confirmatory evidence.

In his discussion of the Laplacian approach, Keynes notes that, "The method is, in fact, much too powerful" (p. 382). In my cursory browsing of Shalizi's web site, that seems to be a criticism that he makes, as well. Keynes mentions that Laplace derived the odds that the sun will rise on the morrow as 1,826,214:1. Keynes continues, "But an ingenious German, Professor Bobek, . . . proves by these same principles that the probability of the sun's rising every day for the next 4,000 years is not more, approximately, than two-thirds" (p. 383).

I was a Bayesian during the time when they were few and far between. I find Keynes attractive because he develops Bayesianism as logic, not math. Another remedy for the deficiencies of the Laplacian approach is that of Good, who posits Type II distributions, which are distributions of Laplacian distributions, Type III distributions, etc. etc. etc. The revival of Bayesianism happened when I was not looking. I have been appalled at statements by some new Bayesians that Cox proved that human cognition was (ideally) Bayesian. Among new Bayesians Pearl sticks out to me as both rigorous and brilliant.

BTW, to my way of thinking, the fact that the sun rose this morning is irrelevant to the question of whether it will rise tomorrow, except for the fact that the one is necessary for the other.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Not quite. I'd repeat the experiment multiple times. If, for example, I were to draw one ball and it were to be black, I'd replace the ball and draw another one. I'd repeat that multiple times to determine the probability of drawing white and black balls, then I would know it some sense what the probability of my next ball being white would be.

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Let's say you did that multiple times, and each time you got a black ball. Then what is the probability of getting a white ball? 0?

I think the math you need to do to solve this is here: http://en.wikipedia.org/wiki/Rule_of_succession (actually this is the math Bill was just talking about...)

If you've done 100 trials and gotten a black go stone each time, then if I did the math right, Laplace says your chance of getting a black stone on the 101st trial is .9902. After 1000 black balls, it's .999. Et cetera...

Special note on probabilities of 0 and 1: if you ever end up with a probability of exactly 0 or 1, you did something wrong, because a probability of that level can no longer change due to new evidence (putting evidence through Bayes' theorem with a prior of 1 or 0 leaves the prior unchanged, no matter how strong the evidence is). See: http://en.wikipedia.org/wiki/Cromwell%27s_rule

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I would not take the bet (for any X$), and I am stronger than 10k.