Cyan comments on Putting in the Numbers - Less Wrong

8 Post author: Manfred 30 January 2014 06:41AM

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Comment author: Oscar_Cunningham 30 January 2014 07:05:26PM 3 points [-]

I'll work in the easier case 1 dimension down. Say we have a die which rolls a 1, 2 or a 3, and we know it averages to 5/2.

Then {x in R^3 : x1+x2+x3=1, xi>=0 for all i} is an equilateral triangle, which we put an uniform distribution on. Then the points where the mean roll is 5/2 lie on a straight line from (1/4,0,3/4) to (0,1/2,1/2). By some kind of linearity argument the averages over this line (with the uniform weighting from our uniform prior) are just the average of (1/4,0,3/4) and (0,1/2,1/2). This gives (1/8,2/8,5/8).

On the other hand we know that maxent gives a geometric sequence. But (1/8,2/8,5/8) isn't geometric.

Comment author: Cyan 30 January 2014 08:03:02PM 3 points [-]

This may help. Abstract:

The method of maximum entropy has been very successful but there are cases where it has either failed or led to paradoxes that have cast doubt on its general legitimacy. My more optimistic assessment is that such failures and paradoxes provide us with valuable learning opportunities to sharpen our skills in the proper way to deploy entropic methods. The central theme of this paper revolves around the different ways in which constraints are used to capture the information that is relevant to a problem. This leads us to focus on four epistemically different types of constraints. I propose that the failure to recognize the distinctions between them is a prime source of errors. I explicitly discuss two examples. One concerns the dangers involved in replacing expected values with sample averages. The other revolves around misunderstanding ignorance. I discuss the Friedman-Shimony paradox as it is manifested in the three-sided die problem and also in its original thermodynamic formulation.

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