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XiXiDu comments on Bayes' Theorem Illustrated (My Way) - Less Wrong

126 Post author: komponisto 03 June 2010 04:40AM

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Comment author: XiXiDu 04 June 2010 09:55:03AM 1 point [-]

Why would you need more than plain English to intuitively grasp Monty-Hall-type problems?

Take the original Monty Hall 'Dilemma'. Just imagine there are two candidates, A and B. A and B both choose the same door. After the moderator picked one door A always stays with his first choice, B always changes his choice to the remaining third door. Now imagine you run this experiment 999 times. What will happen? Because A always stays with his initial choice, he will win 333 cars. But where are the remaining 666 cars? Of course B won them!

Or conduct the experiment with 100 doors. Now let’s say the candidate picks door 8. By rule of the game the moderator now has to open 98 of the remaining 99 doors behind which there is no car. Afterwards there is only one door left besides door 8 that the candidate has chosen. Obviously you would change your decision now! The same should be the case with only 3 doors!

There really is no problem here. You don’t need to simulate this. Your chance of picking the car first time is 1/3 but your chance of choosing a door with a goat behind it, at the beginning, is 2/3. Thus on average, 2/3 of times that you are playing this game you’ll pick a goat at first go. That also means that 2/3 of times that you are playing this game, and by definition pick a goat, the moderator will have to pick the only remaining goat. Because given the laws of the game the moderator knows where the car is and is only allowed to open a door with a goat in it. What does that mean? That on average, at first go, you pick a goat 2/3 of the time and hence the moderator is forced to pick the remaining goat 2/3 of the time. That means 2/3 of the time there is no goat left, only the car is left behind the remaining door. Therefore 2/3 of the time the remaining door has the car.

I don't need fancy visuals or even formulas for this. Do you really?

Comment author: Kaj_Sotala 04 June 2010 04:04:51PM 10 points [-]

I can testify that this isn't anywhere near as obvious to most people than it is to you. I, for one, had to have other people explain it to me the first time I ran into the problem, and even then it took a small while.

Comment author: XiXiDu 05 June 2010 07:00:46PM 1 point [-]

I think the very problem in understanding such issues is shown in your reply. People assume too much, they read too much into things. I never said it has been obvious to me. I asked why you would need more than plain English to understand it and gave some examples on how to describe the problem in an abstract way that might be ample to grasp the problem sufficiently. If you take things more literally and don't come up with options that were never mentioned it would be much easier to understand. Like calling the police in the case of the trolley problem or whatever was never intended to be a rule of a particular game.

Comment author: Kaj_Sotala 06 June 2010 06:50:54AM 2 points [-]

Well, yeah. But if I recall, I did have a plain English explanation of it. There was an article on Wikipedia about it, though since this was at least five years ago, the explanation wasn't as good as it is in today's article. It still did a passing job, though, which wasn't enough for me to get it very quickly.

Comment author: XiXiDu 06 June 2010 09:56:37AM *  2 points [-]

Yesterday, when falling asleep, I remembered that I indeed used the word 'obvious' in what I wrote. Forgot about it, I wrote the plain-English explanation from above earlier in a comment to the article 'Pigeons outperform humans at the Monty Hall Dilemma' and just copied it from there.

Anyway, I doubt it is obvious to anyone the first time. At least anyone who isn't a trained Bayesian. But for me it was enough to read some plain-English (German actually) explanations about it to come to the conclusion that the right solution is obviously right and now also intuitively so.

Maybe the problem is also that most people are simply skeptical to accept a given result. That is, is it really obvious to me now or have I just accepted that it is the right solution, repeated many times to become intuitively fixed? Is 1 + 1 = 2 really obvious? The last page of Russel and Whitehead's proof that 1+1=2 could be found on page 378 of the Principia Mathematica. So is it really obvious or have we simply all, collectively, come to accept this 'axiom' to be right and true?

I haven't had much time lately to get much further with my studies, I'm still struggling with basic Algebra. I have almost no formal education and try to educate myself now. That said, I started to watch a video series lately (The Most IMPORTANT Video You'll Ever See) and was struck when he said that to roughly figure out the doubling time you simply divide 70 by the percentage growth rate. I went to check it myself if it works and later looked it up. Well, it's NOT obvious why this is the case, at least not for me. Not even now that I have read up on the mathematical strict formula. But I'm sure, as I will think about it more, read more proofs and work with it, I'll come to regard it as obviously right. But will it be any more obvious than before? I will simply have collected some evidence for its truth value and its consistency. Things just start to make sense, or we think so because they work and/or are consistent.

Comment author: Blueberry 06 June 2010 05:44:54PM 2 points [-]

But I'm sure, as I will think about it more, read more proofs and work with it, I'll come to regard it as obviously right. But will it be any more obvious than before?

If you're interested, here is a good explanation of the derivation of the formula. I don't think it's obvious, any more than the quadratic formula is obvious: it's just one of those mathematical tricks that you learn and becomes second nature.

Comment author: JoshuaZ 06 June 2010 05:53:45PM *  0 points [-]

I'm not sure I'm completely happy with that explanation. They use the result that ln(1+x) is very close to x when x is small. This is due to the Taylor series expansion of ln(1+x) (edit:or simply on looking at the ratio of the two and using L'Hospital's rule), but if one hasn't had calculus, that claim is going to look like magic.

Comment author: XiXiDu 06 June 2010 10:12:32AM *  2 points [-]

Here are more examples:

Those explanations are really great. I've missed such in school when wondering WHY things behave like they do, when I was only shown HOW to use things to get what I want to do. But what do these explanations really explain. I think they are merely satisfying our idea that there is more to it than meets the eye. We think something is missing. What such explanations really do is to show us that the heuristics really work and that they are consistent on more than one level, they are reasonable.

Comment author: RobinZ 07 June 2010 04:03:25PM *  1 point [-]

That said, I started to watch a video series lately [...] and was struck when he said that to roughly figure out the doubling time you simply divide 70 by the percentage growth rate. I went to check it myself if it works and later looked it up. Well, it's NOT obvious why this is the case, at least not for me. Not even now that I have read up on the mathematical strict formula.

Well, it's an approximation, that's all. Pi is approximately equal to 355/113 - yeah, there's good mathematical reasons for choosing that particular fraction as an approximation, but the accuracy justifies itself. [edited sentence:] You only need one real revelation to not worry about how true Td = 70/r is: that the doubling time is a smooth line - there's no jaggedy peaks randomly in the middle. After that, you can just look how good the fit is and say, "yeah, that works for 0.1 < r < 20 for the accuracy I need".

Comment author: Martin-2 14 February 2013 04:32:33AM *  0 points [-]

Although it's late, I'd like to say that XiXiDu's approach deserves more credit and I think it would have helped me back when I didn't understand this problem. Eliezer's Bayes' Theorem post cites the percentage of doctors who get the breast cancer problem right when it's presented in different but mathematically equivalent forms. The doctors (and I) had an easier time when the problem was presented with quantities (100 out of 10,000 women) than with explicit probabilities (1% of women).

Likewise, thinking about a large number of trials can make the notion of probability easier to visualize in the Monty Hall problem. That's because running those trials and counting your winnings looks like something. The percent chance of winning once does not look like anything. Introducing the competitor was also a great touch since now the cars I don't win are easy to visualize too; that smug bastard has them!

Or you know what? Maybe none of that visualization stuff mattered. Maybe the key sentence is "[Candidate] A always stays with his first choice". If you commit to a certain door then you might as well wear a blindfold from that point forward. Then Monty can open all 3 doors if he likes and it won't bring your chances any closer to 1/2.