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Hook comments on Open Thread: February 2010 - Less Wrong
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I've been reading Probability Theory by E.T. Jaynes and I find myself somewhat stuck on exercise 3.2. I've found ways to approach the problem that seem computationally intractable (at least by hand). It seems like there should be a better solution. Does anyone have a good solution to this exercise, or even better, know of collection of solutions to the exercises in the book?
At this point, if you have a complete solution, I'd certainly settle for vague hints and outlines if you didn't want to type the whole thing. Thanks.
Hint: you need to use the sum rule.
The computation is quite manageable for the case of k=5. For the general case, I too was left feeling dissatisfied with the expression I found, but on reflection I'm somewhat confident it is the correct answer.
The case k=4, Ni=13, m=5 is solved numerically on a Web site which discusses probability for Poker players, that was helpful in checking my results; the answer to 3.2 is a generalization of the results given there.
There does not appear to be a complete collection of solutions. This site comes closest. If I were you I would avoid looking at their solution for exercise 4.1 (I'm trying to forget what little I've seen of it as I'd like to solve 4.1 under my own power), but I would also not feel bad about giving up on 4.1 if you find it difficult.
I'd be happy to discuss Jaynes further over DMs or email - though I may respond at a slow pace, as I'm working through the book as my other activities allow. I'm on chapter 6 now.
Thanks, that was exactly the sort of hint I needed (i.e. of the half dozen different approaches I've been working on, focus on this one). On to 3.3.