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matteri comments on Probability is in the Mind - Less Wrong

60 Post author: Eliezer_Yudkowsky 12 March 2008 04:08AM

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Comment author: matteri 06 May 2011 06:14:22PM *  -2 points [-]

Conrad wrote:

ps - Ofc, knowing, or even just suspecting, the coin is rigged, on the second throw you'd best bet on a repeat of the outcome of the first.

I think it would be worthwhile to examine this conclusion - as it might seem to be an obvious one to a lot of people. Let us assume that there is a very good mechanical arm that makes a completely fair toss of the coin in the opinion of all humans so that we can talk entirely about the bias of the coin.

Let's say that the mechanism makes one toss; all you know is that the coin is biased - not how. Assume that it comes up heads; what does this tell you about the bias? Conrad asserts that it will certainly be biased in favor of heads. How much? Will it always show up as heads? 3 times out of 4? As it turns out, you have no way of knowing.

It could be that it is in fact only 1/3 biased towards heads; then it would be much wiser to bet on tails in the future, no? It could be that it is actually 100 times more likely that tails will come up; you simply can't tell the difference from the first toss.

So let's consider more coin tosses. What if it comes up heads once and then tails 5 times in a row? Could you tell me exactly what the bias is? Is it 5/6 towards tails perhaps? What about 50 tails and 15 heads? In fact, it is still not possible to say anything at all about what the bias is.

Since you probably have a heuristic method of analysis (intuition) you will in time see which side is the best bet; i.e. you'll conclude which side is most likely to be biased and you'll probably be correct - with higher accuracy as the amount of tosses increase. However; there is no logic, rationalism or deduction in the world that could tell you exactly what the bias is. This is true after any integer amount of coin tosses.

Comment author: Alicorn 06 May 2011 06:18:52PM 3 points [-]

It is not necessary to know the exact bias to enact the following reasoning:

"Coins can be rigged to display one face more than the other. If this coin is rigged in this way, then the face I have seen is more likely than the other to be the favored side. If the coin is not rigged in this way, it is probably fair, in which case the side I saw last time is equally likely to come up next by chance. It is therefore a better bet to expect a repeat."

Key phrase: judgment under uncertainty.

Comment author: matteri 06 May 2011 06:24:53PM -1 points [-]

I am not arguing against betting on the side that showed up in the first toss. What is interesting though is that even under those strict conditions, if you don't know the bias beforehand, you never will. Considering this; how could anyone ever argue that there are known probabilities in the world where no such strict conditions apply?

Comment author: Alicorn 06 May 2011 06:30:09PM 0 points [-]
Comment author: matteri 06 May 2011 06:43:23PM 0 points [-]

Very well, I could have phrased it in a better way. Let me try again; and let's hope I am not mistaken.

Considering that even if there is such a thing as an objective probability, it can be shown that such information is impossible to acquire (impossible to falsify); how could it be anything but religion to believe in such a thing?

Comment author: Alicorn 06 May 2011 06:47:42PM 0 points [-]
Comment author: soreff 06 May 2011 06:23:31PM *  1 point [-]

However; there is no logic, rationalism or deduction in the world that could tell you exactly what the bias is. This is true after any integer amount of coin tosses.

This seems like it is asking too much of the results of the coin tosses. Given some prior for the probability distribution of biased coins, each toss result updates the probability distribution. Given a prior probability distribution which isn't too extreme (e.g. no zeros in the distribution), after enough toss results, the posterior distribution will narrow towards the observed frequencies of heads and tails.

Yes, at no point is the exact bias known. The distribution doesn't narrow to a delta function with a finite number of observations. So?