In Defense of Objective Bayesianism: MaxEnt Puzzle.
In Defense of Objective Bayesianism by Jon Williamson was mentioned recently in a post by lukeprog as the sort of book that should be being read by people on Less Wrong. Now, I have been reading it, and found some of it quite bizarre. This point in particular seems obviously false. If it’s just me, I’ll be glad to be enlightened as to what was meant. If collectively we don’t understand, that’d be pretty strong evidence that we should read more academic Bayesian stuff.
Williamson advocates use of the Maximum Entropy Principle. In short, you should take account of the limits placed on your probability by the empirical evidence, and then choose a probability distribution closest to uniform that satisfies those constraints.
So, if asked to assign a probability to an arbitrary A, you’d say p = 0.5. But if you were given evidence in the form of some constraints on p, say that p ≥ 0.8, you’d set p = 0.8, as that was the new entropy-maximising level. Constraints are restricted to Affine constraints. I found this somewhat counter-intuitive already, but I do follow what he means.
But now for the confusing bit. I quote directly;
“Suppose A is ‘Peterson is a Swede’, B is ‘Peterson is a Norwegian’, C is ‘Peterson is a Scandinavian’, and ε is ‘80% of all Scandinavians are Swedes’. Initially, the agent sets P(A) = 0.2, P(B) = 0.8, P(C) = 1 P(ε) = 0.2, P(A & ε) = P(B & ε) = 0.1. All these degrees of belief satisfy the norms of subjectivism. Updating by maxent on learning ε, the agent believes Peterson is a Swede to degree 0.8, which seems quite right. On the other hand, updating by conditionalizing on ε leads to a degree of belief of 0.5 that Peterson is a Swede, which is quite wrong. Thus, we see that maxent is to be preferred to conditionalization in this kind of example because the conditionalization update does not satisfy the new constraints X’, while the maxent update does.”
p80, 2010 edition. Note that this example is actually from Bacchus et al (1990), but Williamson quotes approvingly.
His calculation for the Bayesian update is correct; you do get 0.5. What’s more, this seems to be intuitively the right answer; the update has caused you to ‘zoom in’ on the probability mass assigned to ε, while maintaining relative proportions inside it.
As far as I can see, you get 0.8 only if we assume that Peterson is a randomly chosen Scandinavian. But if that were true, the prior given is bizarre. If he was a randomly chosen individual, the prior should have been something like P(A & ε) = 0.16 P(B & ε) = 0.04 The only way I can make sense of the prior is if constraints simply “don’t apply” until they have p=1.
Can anyone explain the reasoning behind a posterior probability of 0.8?
I contacted Williamson about this, and he wrote back:
Ahhh. Thanks for emailing, and for his relying.
It seems like you could just say, “The odds of a coin being heads are 1⁄2. However, subjectivism allows you to say they’re 1 (or .9999 at any rate). Hence, subjectivism is wrong.”
This bit doesn’t make sense either, what kind of evidence imposes a condition on your personal (subjective) probability?
A subjective probability is not arbitrary. It is the most accurate estimate possible given the evidence to the subject. See http://lesswrong.com/lw/s6/probability_is_subjectively_objective .
I think you’re misunderstanding me, but I can’t think of a way to better phrase what I said, sorry.
Maybe if I put it thus: I can’t think of a situation where some evidence E will make my posterior probability P(A|E) greater than 0.8 regardless of my prior P(A).
Yeah, re-reading the quote, I see what you mean. He seems to have confused a frequency with a probability distribution over possible values of the frequency. Maybe that’s why he made the other error that the post discusses.
Are there other examples that have the same bizarrely wrong results?
There are a couple of others, all of which seem to rely on being told probabilities. You assign P(A) = 0.5, and then get told that P(A) = 0.7.
It seems that either P(A) = 0.7 is someone else’s degree of belief, in which case Aumann comes into play, or a statement about what your degree of belief should be, given your evidence. But idealised Bayesian agents don’t make that sort of mistake!
The prior given implies there was other evidence that Peterson was a Swede and Williamson wants to ignore it, but doesn’t explain why (that you’ve quoted anyway).
I quoted the entire section. I suppose I could see if there’s any more in Bacchus.
This seems wrong to me too. His prior shows that he has some other information about Peterson, but then he throws it out.
P(A|ε) = P(A & ε)/P(ε) = 0.1/0.2 = 0.5 != 0.8
This failure to use all of the evidence is exactly what Jaynes warns against for much of PT:LoS.