We throw it many times (say, a billion) and obtain an average value of 3.5. Using that information alone
So my prior state of knowledge about the die is entirely characterized by N=10^9 and m=3.5, with no knowledge of the shape of the distribution? It’s not obvious to me how you’re supposed to turn that, plus your background knowledge about what sort of object a die is, into a prior distribution; even one that maximizes entropy. The linked article mentions a “constraint rule” which seems to be an additional thing.
This sort of thing is rather thoroughly covered by Jaynes in PT:TLOS as I recall, and could make a good exercise for the Book Club when we come to the relevant chapters. In particular section 10.3 “How to cheat at coin and die tossing” contains the following caveat:
The results of tossing a die many times do not tell us any definite number char-
acteristic only of the die. They tell us also something about how the die was tossed. If
you toss ‘loaded’ dice in different ways, you can easily alter the relative frequencies of
the faces. With only slightly more difficulty, you can still do this if your dice are perfectly
‘honest’.
And later:
The problems in which intuition compels us most strongly to a uniform probability
assignment are not the ones in which we merely apply a principle of ‘equal distribution
of ignorance’. Thus, to explain the assignment of equal probabilities to heads and tails on the grounds that we ‘saw no reason why either face should be more likely than the other’, fails utterly to do justice to the reasoning involved. The point is that we have not merely ‘equal ignorance’. We also have positive knowledge of the symmetry of the problem; and introspection will show that when this positive knowledge is lacking, so also is our intuitive compulsion toward a uniform distribution.
Hah. The dice example and the application of maxent to it comes originally from Jaynes himself, see page 4 of the linked paper.
I’ll try to reformulate the problem without the constraint rule, to clear matters up or maybe confuse them even more. Imagine that, instead of you throwing the die a billion times and obtaining a mean of 3.5, a truthful deity told you that the mean was 3.5. First question: do you think the maxent solution in that case is valid, for some meaning of “valid”? Second question: why do you think it disagrees with Bayesian updating as you throw the die a huge number of times and learn only the mean? Is the information you receive somehow different in quality? Third question: which answer is actually correct, and what does “correct” mean here?
I’m not really qualified to comment on the methodological issues since I have yet to work through the formal meaning of “maximum entropy” approaches. What I know at this stage is the general argument for justifying priors, i.e. that they should in some manner reflect your actual state of knowledge (or uncertainty), rather than be tainted by preconceptions.
If you appeal to intuitions involving a particular physical object (a die) and simultaneously pick a particular mathematical object (the uniform prior) without making a solid case that the latter is our best representation the former, I won’t be overly surprised at some apparently absurd result.
It’s not clear to me for instance what we take a “possibly biased die” to be. Suppose I have a model that a cubic die is made biased by injecting a very small but very dense object at a particular (x,y,z) coordinate in a cubic volume. Now I can reason based on a prior distribution for (x,y,z) and what probability theory can possibly tell me about the posterior distribution, given a number of throws with a certain mean.
Now a six-sided die is normally symmetrical in such a way that 3 and 4 are on opposite sides, and I’m having trouble even seeing how a die could be biased “towards 3 and 4” under such conditions. Which means a prior which makes that a more likely outcome than a fair die should probably be ruled out by our formalization—or we should also model our uncertainty over how which faces have which numbers.
I’m having trouble even seeing how a die could be biased “towards 3 and 4” under such conditions.
If the die is slightly shorter along the 3-4 axis than along the 1-6 and 2-5 axes, then the 3 and 4 faces will have slightly greater surface area than the other faces.
Our models differ, then: I was assuming a strictly cubic die. So maybe we should also model our uncertainty over the dimensions of the (parallelepipedic) die.
But it seems in any case that we are circling back to the question of model checking, via the requirement that we should first be clear about what our uncertainty is about.
So my prior state of knowledge about the die is entirely characterized by N=10^9 and m=3.5, with no knowledge of the shape of the distribution? It’s not obvious to me how you’re supposed to turn that, plus your background knowledge about what sort of object a die is, into a prior distribution; even one that maximizes entropy. The linked article mentions a “constraint rule” which seems to be an additional thing.
This sort of thing is rather thoroughly covered by Jaynes in PT:TLOS as I recall, and could make a good exercise for the Book Club when we come to the relevant chapters. In particular section 10.3 “How to cheat at coin and die tossing” contains the following caveat:
And later:
Hah. The dice example and the application of maxent to it comes originally from Jaynes himself, see page 4 of the linked paper.
I’ll try to reformulate the problem without the constraint rule, to clear matters up or maybe confuse them even more. Imagine that, instead of you throwing the die a billion times and obtaining a mean of 3.5, a truthful deity told you that the mean was 3.5. First question: do you think the maxent solution in that case is valid, for some meaning of “valid”? Second question: why do you think it disagrees with Bayesian updating as you throw the die a huge number of times and learn only the mean? Is the information you receive somehow different in quality? Third question: which answer is actually correct, and what does “correct” mean here?
I think I’d answer, “the mean of what?” ;)
I’m not really qualified to comment on the methodological issues since I have yet to work through the formal meaning of “maximum entropy” approaches. What I know at this stage is the general argument for justifying priors, i.e. that they should in some manner reflect your actual state of knowledge (or uncertainty), rather than be tainted by preconceptions.
If you appeal to intuitions involving a particular physical object (a die) and simultaneously pick a particular mathematical object (the uniform prior) without making a solid case that the latter is our best representation the former, I won’t be overly surprised at some apparently absurd result.
It’s not clear to me for instance what we take a “possibly biased die” to be. Suppose I have a model that a cubic die is made biased by injecting a very small but very dense object at a particular (x,y,z) coordinate in a cubic volume. Now I can reason based on a prior distribution for (x,y,z) and what probability theory can possibly tell me about the posterior distribution, given a number of throws with a certain mean.
Now a six-sided die is normally symmetrical in such a way that 3 and 4 are on opposite sides, and I’m having trouble even seeing how a die could be biased “towards 3 and 4” under such conditions. Which means a prior which makes that a more likely outcome than a fair die should probably be ruled out by our formalization—or we should also model our uncertainty over how which faces have which numbers.
If the die is slightly shorter along the 3-4 axis than along the 1-6 and 2-5 axes, then the 3 and 4 faces will have slightly greater surface area than the other faces.
Our models differ, then: I was assuming a strictly cubic die. So maybe we should also model our uncertainty over the dimensions of the (parallelepipedic) die.
But it seems in any case that we are circling back to the question of model checking, via the requirement that we should first be clear about what our uncertainty is about.
Cyan, I was hoping you’d show up. What do you think about this whole mess?
I find myself at a loss to give a brief answer. Can you ask a more specific question?