Timeline for Coin flipping, decision processes and value of information
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 1, 2013 at 19:14 | history | undeleted | whuber♦ | ||
| Mar 15, 2013 at 21:53 | history | deleted | Douglas Zare | ||
| Mar 15, 2013 at 21:47 | comment | added | whuber♦ | Douglas: Because I paid more attention to your answer :-). Please don't get me wrong--I like it and I like this thread. I thought it important to point out that you had to add an assumption in order to obtain your answer, that's all. As a practical matter, in many situations--including this one--there is no prior. (I sure wouldn't want to make up a personal prior and then have to bet big money on it!) But of course there is still an optimum, provided you specify a loss function. ("Maximizing" an expectation is not a full loss function.) | |
| Mar 15, 2013 at 21:35 | comment | added | Douglas Zare | @M. Cypher: The uniform prior seemed simplest. What's the advantage to using a Jeffreys prior? As I stated before, the same technique works for other priors, too. | |
| Mar 15, 2013 at 21:22 | comment | added | M. Cypher | @Douglas: I'm wondering, is there a reason you used the uniform prior and not Jeffrey's prior? | |
| Mar 15, 2013 at 20:56 | comment | added | Douglas Zare | @whuber: Also, why not make the same objection to Cam Davidson Pilon's answer? It seems people like that one better even though it's clearly wrong, so I feel like deleting this one. | |
| Mar 15, 2013 at 20:51 | comment | added | Douglas Zare | @jerad: It's just dynamic programming with memoization en.wikipedia.org/wiki/Memoization. I computed the expected number of heads from the best strategy (the equity) as the maximum over the expected number of heads over each possible option at the step. | |
| Mar 15, 2013 at 20:47 | comment | added | Douglas Zare | @whuber: Do you think I made a mistake in assuming there might be a prior distribution, and stating which prior I was using? The same "mistake" seems to be made by every paper I can find on the subject. Without a prior, I'm not sure how you can optimize anything. | |
| Mar 15, 2013 at 19:27 | comment | added | jerad | I don't know Mathematica so I can't follow how you computed your expected number of heads. Care to explain that part? If we assume knowledge that coin B's bias is drawn from a uniform distribution on [0,1], then I don't see how you can expect to beat 50/50. | |
| Mar 15, 2013 at 19:06 | comment | added | whuber♦ | What permits us to assume there is a prior distribution on B? I admit that such an assumption makes the problem more tractable, but the existence of an objectively valid assessment of the fairness of B is doubtful to me. Yes, we do have the results of some previous flips, but those are still consistent with any value for $\Pr_B(\text{heads})$ in $(0,1)$. If in fact that probability is less than $1/2$, then I don't care what prior you choose to adopt: it will be an objective fact that the expected number of heads with your approach is less than $50$. | |
| Mar 15, 2013 at 16:33 | comment | added | Douglas Zare | Thompson sampling is a poor approximation. There are better approximations you can use if you don't want to go through the trouble of the (at worst quadratic) exact calculation, but still want to avoid large errors. Actually, the exact calculation might be closer to linear. | |
| Mar 15, 2013 at 16:29 | history | edited | Douglas Zare | CC BY-SA 3.0 |
Added comparison with Thompson sampling
|
| Mar 15, 2013 at 16:09 | comment | added | M. Cypher | I agree that an optimal solution would be better than an approximate one. I wonder if there is an optimal general solution which can be efficiently applied within milliseconds in a dynamic environment with several hundred "coins". If not, I guess Thompson sampling is the best option. | |
| Mar 15, 2013 at 15:08 | history | answered | Douglas Zare | CC BY-SA 3.0 |