Over at Equitable Growth: Apropos of:
Cosmos Elysée (2009): On the Certainty of the Bayesian Fortune-Teller,
- Sky Masterson: An Ear Full of Cider,
- Adam Elga (2010): Subjective Probabilities Should Be Sharp, and
- Brad DeLong: [Tuesday Virtual Office Hours: Follow-Up Questions on the Philosophy of Probability5...
Thrasymakhos: Today we discover that Sam Wang does not seem to be a Bayesian:
Sam Wang: Why you’re wrong to get excited about “60%”: Some people are excited... Nate Silver... [gives] a probability of a GOP [Senate] takeover at 60%. To cut to the chase: I do not think that number means what you think it does...
Thomas Bayes: It is simple. It means that Nate Silver stands ready to bet on Senate control next January at odds of 3-2.
Thrasymakhos: "Stands ready"? READ MOAR
Thomas Bayes: Yes. He stands ready to make a (small) bet that the Majority Leader of the Senate will be a Republican on January 5, 2015 if he gets at least 2-3 odds, and he stands ready to make a (small) bet that the Majority Leader of the Senate will be a Republican on January 5, 2015 if he gets at least 3-2 odds. Since Nate Silver has gained considerable success so far in life by making predictions and laying odds that reality has thereafter validated, his views on the odds are worthy of great respect unless you think you have important private information that he does not or a superior analytical methodology--and you probably should not think that, as those who did think they had better ideas of the odds than Nate Silver in the past are, for the most part, shirtless. What else could it possibly mean? What else could people take it to mean?
Jerzey Neyman: No, no, no! You have got it all wrong! Sam Wang has it right!
Sam Wang: Think of five... tosses... [of] coins are not perfectly fair, and the overall situation is a little unfavorable to Democrats. That is basically the amount of uncertainty expressed in Silver’s probability. Fundamentally, any probability in the 40-60% range is a numerical way of saying “I don’t know.”.... The certainty fallacy. Silver has done something common among paid writers, which is to do what it takes to attract eyeballs. He has rounded a probability that is barely over 50% to make the statement that one side is ahead.... Basically, whenever you see a probability like that, you should mentally say “plus or minus 20%” just to get the right idea...
Thomas Bayes: Now I am confused. So Sam Wang thinks:
- Nate Silver should say that the probability of a GDP Senate takeover might be 40%, might be 60%, and might be 80%-that would give people the right idea.
- Nate Silver should not say that the probability right now, August 25, 2014, with 71 days to go before the election, of a GDP Senate takeover is 60%--that would give people the wrong idea.
- Nate Silver should not say that the probability of a GDP Senate takeover might be 30%, might be 60%, and might be 90%--that would give people the wrong idea.
- Nate Silver should not say that the probability of a GDP Senate takeover might be 50%, might be 60%, and might be 70%--that would give people the wrong idea.
Do I have it right?
Jerzey Neyman: Exactly!
Thomas Bayes: May I say that I am having a much harder time understanding what Sam Wang means--when he says that we should say that the probability is in [0.4, 0.8], and not say that the probability is 0.6 or that it is in [0.3, 0.9] or in [0.5, 0.7]--than what Nate Silver means when he says that he thinks the probability is 0.6?
Jerzey Neyman: You may say it. But why should that be hard to understand?
Thomas Bayes: I find it hard to understand how a probability could be something different than it currently is.
Jerzey Neyman: But surely you agree that somebody who knew more than you did about the forthcoming Senate election would have a different estimated probability of Republican takeover than your and Nate Silver's 60%? And that his or her estimate would be a better probability than yours?
Thomas Bayes: Yes, of course. Probabilities are associated with information sets. As information arrives and information sets grow...
Sky Masterson: Including, importantly, growing by adding the information that somebody with an information set much larger than yours wants to make a large bet against against you at your probability...
Thomas Bayes: ...the probability you hold shifts. That's what "learning stuff" means.
Jerzey Neyman: Maybe I can make an analogy that will help you understand. Suppose that there were a large urn full of marbles--red marbles and blue marbles. Suppose that there is a being, named Tykhe, who periodically takes marbles out of the urn at random by a process that will leave only one marble in the urn on November 4, 2014. Suppose that if that one remaining marble is red then a Republican becomes majority leader of the Senate in January 2015, and if that one remaining marble is blue then a Democrat becomes majority leader of the Senate in January 2015. Suppose we know right now that there are 100 marbles left in Tykhe's urn, and that somewhere between 40 and 80 of them are red. Thus we should say that the probability of Republican senate control in the next election is between 40% and 80%. We should not say that the probability is 60% because our knowledge of the urn does not extend to knowing that there are 60 red marbles in it. We should not say that the probability is between 50% and 70% because we do not know that there are not 45 or 75 red marbles in it. And we should not say that the probability is between 30% and 90% because we do know that there are not 30 red marbles in it and not 90 red marbles in it. Is that clear?
Thomas Bayes: But there is no urn...
Thrasymakhos: So, Jerzey and Sam, your position is that we should reserve the word "probability" and use it only to refer to the betting odds of somebody with the superior information set provided by being able to look into Tykhe's urn right now and count the marbles?
Jerzey Neyman: Exactly!
Thomas Bayes: But there is no urn. And there is no being Tykhe...
Thrasymakhos: It does make me wonder. Why do you reserve "probability" for the betting odds that someone who had the information set associated with having counted today's--and not yesterday's, and not tomorrow's, and not last year's, and not November 4's--marbles in Tykhe's urn would offer, and not the betting odds of somebody with a different information set?
Jerzey Neyman: The urn and Tykhe are an analogy. The point of the urn and Tykhe is that the real probabilities are those associated with an observer who understands the generating process, and whose uncertainty is over (a) how exactly some of the details of that generating process have played out to date and (b) the effect of unknowable future events in the generating process.
Thomas Bayes: But there is no urn. And there is no being Tykhe. And there are no marbles...
Mentor: Ah! It is always interesting to see young sophonts with poorly because freshly-evolved brains of a low order of intelligence attempt to wrestle with these conundrums. I am Mentor, of Arisia, an anthology intelligence of a high order. My visualization of the Cosmic All is correct to a tolerance of 2^(-50)--and with a confidence of 2^(-50) I know who will control the senate come January 2015. What you see as unknown details of how the generating process has played out to date are things I know as well as I would know the back of my own hand, were I the kind of being that had hands. What you see as the effect of unknowable future events I can foresee as easily as you can predict how old you will be the next time your birthday falls on a Sunday--or... no, I decided 10,000,000 years ago I would not tell you that and cut off this explanation here...
Thrasymakhos: Seems to me, Sam and Jerzey, that as long as creatures like Mentor exist, all probabilities must be either 2^(-50) or 1-2^(-50).
Jerzey Neyman: But no such creatures as Mentor exist!
Thomas Bayes: But there is no urn. And there is no being Tykhe. And there are no marbles. There are no red marbles. There are no blue marbles...
Mentor: I beg your pardon?
Jerzey Neyman: You are a fictional character in a space opera, not a real being!
Mentor: And what do you suppose you are? Do you think you are real?
Thomas Bayes: But there is no urn. And there is no being Tykhe. And there are no marbles. There are no red marbles. There are no blue marbles. How can the limits of our knowledge about marbles that do not exist in an urn that does not exist manipulated by a being that does not exist have any impact on our probabilities of things that do exist?...
Mentor: I am at least a much-beloved figure in space operas that have had a wide readership for three generations.
Sokrates: Would Sam Wang accept, as a friendly amendment, the proposition that Nate Silver should say not that the probability is between 40% and 80% right now, but that if he could access and fully process all the information that is already out there today it might push the probability down to 40%, it might push it up to 80%, it might keep it the same?
Thrasymakhos: So Mentor's assessments cannot be probabilities because he doesn't exist...
Thomas Bayes: But Nate Silver can't. Nobody can.
Mentor: I can! And more...
Thrasymakhos: Yet the assessments of the nonexistent person looking at nonexistent Tykhe's nonexistent urn are probabilities even though none of the three exist?
Jerzey Neyman: It's just an analogy. Real world probabilities are much more like an unknown but bounded number of different kinds of marbles in urns than they are like Bayesian or Arisian woo-woo!
Sokrates: But if he could...
Thomas Bayes: But he can't. And because he can't, and doesn't know whether doing something he can't do would push it down to 40%, up to 80%, or leave it unchanged, Nate Silver should say that the probability is 60%