Botanizing Probability

by Ian Pollock

A bearded stranger approaches you on the street and starts pulling objects out of a sack. The first three objects he pulls out are (1) a large tube of toothpaste, (2) a live jellyfish, (3) a small tube of toothpaste. Now, what is your level of credence that the next object pulled out of the bag is a tube of toothpaste? How much are you willing to bet on toothpaste, and at what odds?

This is how Adam Elga begins his excellent paper, subjective probabilities should be sharp, and it seems a fitting introduction to an odd little problem in philosophy of probability. [1] What are we to make of a situation like the above?

As Elga argues in his paper, we might start by trying to make a distinction between three relevant situations in which a probability judgment is requested of us. The evidence can be, in his words, sharp (for example, the probability that it will be 20°C or higher in Winnipeg on March 15, for which a farmer’s almanac or weather forecast should be just fine), or sparse but with a clear upshot (for example, the probability that the next person in the UK to be named Suzanne will be born at an odd hour of the day — we have no idea when this birth will take place, but 50% is the way to bet).

But sometimes, as in  the above example of the jellyfish/toothpaste draw, evidence is sparse and unspecific. Assigning a probability to toothpaste would require theorizing about the stranger’s motives as well as the bag’s contents. But both are utterly mysterious to us. There is a temptation here to refuse to make a probability judgment, or to say that the probability is “vague” or “in some interval.” Elga’s paper goes on to argue that this last proposal is not really acceptable (it ends up implying that you should reject a Dutch book — a sequence of bets guaranteed to make you money no matter what the outcome).

This is only the most extreme example of the strange effects of “vagueness” on probability judgments. In a classic paper, Daniel Ellsberg goes over some other situations, such as the following:

Urn A contains exactly 50 red balls and 50 black balls.
Urn B contains 100 red or black balls, but you don’t know the relative quantities. It might be 50-50, 0 red and 100 black, 100 red and 0 black, or anything in between.
You are offered two tickets. Ticket A pays $100 if a red is drawn from Urn A. Ticket B pays $100 if a red is drawn from Urn B. Which would you be willing to pay the most for?

This is one instantiation of the Ellsberg paradox. A moment’s thought should show that your probability on drawing red should be 50% (1:1 odds) in both cases. But we see that there is an intuitive tendency to prefer the wager in which the “dynamics” of the problem are known (Ticket A).

But wait! Suppose we keep everything the same, but Tickets A and B pay out if a black ball is drawn. Symmetrically, Ticket A would still be preferred to Ticket B, since Ticket A is less “vague.” But then your inferred probability for “red from Urn B” must be less than “red from Urn A,” while at the same time, “black from Urn B” is less than “black from Urn A.” Since the probabilities from Urn A must sum to 100%, this means your probabilities from Urn B are summing to less than 100%. This earns you a special place in the 3rd circle of Bayes Hell.

So this preference for “non-vague” bets leads to absurdities if we attempt to interpret your betting behavior as probability. This is a problem, because many philosophers, most notably Frank Ramsey, have wanted to enshrine “whatever determines betting preferences” as a sort of operational, extensive definition of probability.

My question is fairly broad: how can we make sense of what is going on in cases like these?

Let’s start with the Ellsberg paradox, as it seems like a more tractable case.

First, I will declare an interest: I do not share the intuition that prefers the known urn to the unknown one — I am completely indifferent between the two bets. That being said, I have a few ideas about why some people might prefer the bet in which the urn contents are known. Ranked in order of plausibility (although they are not mutually exclusive), they are:

1. You would feel stupid if you chose Ticket B (the unknown urn) and it turned out that there were only black balls in Urn B. You have a strong preference not to feel stupid, so you’re willing to pay more for Urn A, which is guaranteed to be a “fair” urn. [2]
2. You have been conditioned to think of a probability as a property of a situation, rather than a property of an epistemic state. (This is encouraged by our conventions of language, as in “the probability of rain tomorrow is 20%,” which uses the definite article “the” and thus implies a single uniquely correct value of probability, independent of what anybody knows about it. I prefer to phrase these things as “I give odds of 4:1 against rain tomorrow.”) For this reason, you feel that you do not know the probabilities for Urn B, and so you do not wish to bet on it.
3. You are wary of being tricked by whoever is holding the draw into betting on a lame horse. An urn with unknown quantities of red and black seems like a potential trick.

If I am right, then in my opinion only one of these objections is defensible (number 3). But they do seem to me to do a half-decent job of explaining away this intuition.

But what about the toothpaste/jellyfish draw? Here, I do share the feeling that says there may be no uniquely correct way to proceed. However, I think I may be able to demystify where that intuition is coming from, and what we should make of it. Give it the old college try, anyway.

For me, the interesting thing about the toothpaste/jellyfish draw is that my thought process when I consider it is very unstable. My stream of consciousness looks something like this:

“Okay Ian, two toothpaste tubes and one jellyfish. What do we expect next?”
“I don’t know. How long is a piece of string?”
“What about Laplace’s rule of succession? (s+1)/(n+2). Defining success as toothpaste and non-success as non-toothpaste, we get (2+1)/(3+2)=3/5 probability of toothpaste.”
“Yeah, I’m not sure that that is even applic - SQUIRREL!”
“Pay attention! How much would you be willing to pay for a ticket that paid out $10 if toothpaste was drawn?”
“I dunno, maybe I’d give $2.”
“Okay, so that implies your odds are 4:1 against toothpaste.”
“I think that reflects the triumph of curiosity over thrift, more than it does any real probability judgment. I would not pay $200 for a $1000 ticket. Or maybe I would.”
“Look, just answer this: how likely is toothpaste? You can see that 2 out of 3 things pulled out of the bag have been toothpaste. That is evidence that toothpaste is common in the bag.”
“If you say so. Do we even know that this is a random draw? Maybe the guy draws whatever he wants to, and that depends on how I bet. Why is he even performing this draw? He’s probably trying to trick me.”
“Trick you into betting for toothpaste, or against toothpaste? By the way, what’s wrong with the Laplace’s rule approach, again?”
“I don’t know. Please go boil your head.”

Clearly, if one’s probability judgments are jumping all over the place from second to second and minute to minute, the smart thing to do is to refuse to either quote them or act upon them until they settle into a stable range (if they do so at all). It’s not that our probabilities are “vague” or “unsharp” in this situation. It’s that we are unable to assign them at all due to our severe epistemic limitations. “Vague” describes the phenomenology of such a situation (we feel a sense of confusion about how to proceed), rather than the actual answer one should give.

I think I can get even more specific on this. Consider that there are two types of uncertainty we encounter in our lives: empirical uncertainty, and logical uncertainty. Empirical uncertainty is not knowing who won the Superbowl. Logical uncertainty is not knowing the cube root of 74,088.

The important thing to notice about logical uncertainty is that it is relative to our abilities to draw deductions from knowledge in a timely way, rather than relative to our knowledge itself. The most brilliant mind ever to have lived can sit in an armchair all day, and not be able to tell you who won the Superbowl. But they ought to be able to tell you the cube root of 74,088, given enough time. In fact, I should be able to as well. The cube root of 74,088 is *implied knowledge*, like most of mathematics and some of philosophy. In a sense, you already know it, because you know other things which logically entail it.

What’s the probability that the cube root of 74,088 is greater than 35? This is a strange question precisely because there is no empirical uncertainty about the answer at all. But probabilistic reasoning only works well with empirical uncertainty, not logical uncertainty. Actually, correct use of Bayes rule in a sense assumes “logical omniscience,” wherein you are perfectly able to comprehend the logical implications of all hypotheses under consideration.

Thus, my tentative resolution of the jellyfish/toothpaste paradox is that its paradoxical nature appears because we are not logically omniscient, and the situation into which the thought experiment places us is both empirically and logically intractable. It is as though our inner calculator had a stack overflow. We are used to probability judgments under empirical uncertainty but logical certainty — for example, the chance of getting tails on a coin flip. The introduction of severe logical uncertainty makes us reluctant to try to calculate any probabilities at all (we just don’t know where to start analyzing the situation) — hence the intuitive appeal of an evasive answer to the jellyfish/toothpaste draw.

Does this seem satisfying to you?

_________

[1] I first heard about this problem, however, from the Phi2Phi app.

[2] E.T. Jaynes tells the story of an objection to the US draft lottery; according to the plaintiff, the drawing was not “truly random” because the bowl containing names had not been sufficiently mixed. Asks Jaynes, “To whom is it unfair?”