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

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

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Comment author: TaeKahn 08 June 2010 02:05:47PM -1 points [-]

I have a small issue with the way you presented the Monty python problem. In my opinion, the setup could be a little clearer. The Bayesian model you presented holds true iff you make an assumption about the door you picked; either goat (better) or car (less wrong). If you pick a door at random with no presuppositions (I believe this is the state most people are in), then you have no basis to decide to switch or not switch, and have a truly 50% chance either way. If instead you introduce the assumption of goat, when the host opens up the other goat, you know you had a 2/3 chance to pick a goat. With both goats known or presumed, the last door must be the car with an error rate of 1/3.

Comment author: RobinZ 08 June 2010 03:23:00PM 0 points [-]

As far as I can see, one-third of the time the first door you picked had the car. What happens afterward cannot change that one-third. The only way it could change your one-third credence is if sometimes Monty Hall did one thing and sometimes another, depending on whether you picked the car.

Comment author: TaeKahn 08 June 2010 05:39:55PM *  0 points [-]

While the overall probabilities for the game will never change, the contestant’s perception of the current state of the game will cause them to affect their win rate. To elaborate on what I was saying, imagine the following internal monologue of a contestant:

I’ve eliminated one goat. Two doors left. One is a goat, one is a car. No way to tell which is which, so I’ll just randomly pick one.

IMO, this is probably what most contestants believe when faced with the final choice. Obviously, there is a way to have a greater success rate, but this person is evaluating in a vacuum. If contestants were aware of the actual probabilities involved, I think we would see less “agonizing” moments as the contestants decide if they should switch or not. By randomly picking door A or B, irrespective of the entirety game, you’ve lost your marginal advantage and lowered your win rate. That being said, if they still “randomly” pick switch every their win rate will be the expected, actual probability.

Edit: The same behaviour can be seen in Deal or No Deal. If for some insane reason, they go all the way to the final two cases, the correct choice is to switch. I don’t know exactly how many cases you have to choose from, but the odds are greatly against you that you picked the 1k case. If the case is still on the board, the chick is holding it. Yet, people make the choice to switch based entirely on the fact that there are two cases and 1 has 1k and the other has 0.01. They think they have a 50/50 shot, so they make their odds essentially 50/50 by randomly choosing. In other words, they might as well as have flipped a coin to make the decision between the two cases.

Comment author: Jack 09 June 2010 03:06:14PM *  6 points [-]

Actually, I just realized... there is no reason to swap on Deal or No Deal. The reason why you swap in Monty Hall is that Monty knows which door has the goats and there is no chance he will open a door to reveal a car. But in Deal or No Deal the cases that get opened are chosen by the contestant with no knowledge of what is inside them. It's like if the contestant got to pick which of the two remaining doors to open instead of Monty, there is a 1/3 chance the contestant would open the door with the car leaving her with only goats to choose from. The fact the the contestant got lucky and didn't open the door with the car wouldn't tell her anything about which of the two remaining doors the car is really behind.

ETA: Basically Deal or No Deal is just a really boring game.

Comment author: thomblake 09 June 2010 03:26:29PM 0 points [-]

Basically Deal or No Deal is just a really boring game.

Well, it's exciting for those who like high-stakes randomness. And there are expected utility considerations at every opportunity for a deal (I don't remember if there's a consistent best choice based on the typical deal).

Comment author: Jack 09 June 2010 03:29:15PM 1 point [-]

And there are expected utility considerations at every opportunity for a deal (I don't remember if there's a consistent best choice based on the typical deal).

I was talking about this in my other comment.

Comment author: gwern 09 June 2010 05:44:52PM 0 points [-]

Maybe it could be interesting if you treat it as a psychology game - trying to predict, based on the person's appearance, body language, and statements, whether they will conform to expected-utility or not?

Comment author: Jack 08 June 2010 08:00:57PM *  3 points [-]

If for some insane reason, they go all the way to the final two cases

The way the deals work going down to the final two cases can end up the best strategy. Basically they weight the deals to encourage higher ratings. As long as there is a big money case in play they won't offer the contestant the full average of the cases- presumably viewers like to watch people play for the big money so the show wants these contestants to keep going. If all the big money cases get used up the banker immediately offers the contestant way more than they are worth to get them off the stage and make way for a new contestant.

(This was my conclusion after watching until I thought I had figured it out. I could be reading this into a more complex or more random pattern.)

Comment author: JoshuaZ 08 June 2010 06:04:13PM 2 points [-]

Actually, people are much more likely to not switch if they think they are both equal. I've just finished reading Jason Rosenhouse's excellent book on the Monty Hall Problem and there's a strong tendency for people not to switch. Apparently people are worried that they'll feel bad if they switch and that action causes them to lose. (The book does a very good job discussing a lot about both the problem and psych studies about how people react to it or different versions. I strongly recommend it.)

Comment author: RobinZ 08 June 2010 05:43:51PM *  1 point [-]

On the actual show, sometimes Monty Hall did one thing and sometimes another, so far as I am told. We're not talking about actual behavior of contestants in an actual contest, we're talking about optimal behavior of contestants in a fictional contest.

Edit: I'm sorry, I really don't know what you're arguing. Am I making sense?

Comment author: TaeKahn 08 June 2010 07:21:55PM 0 points [-]

You are making perfect sense; it’s me that is not. I had thought to clarify the issue for people that might still not “get it” after reading the article. Instead, I’ve only muddied the waters.

Comment author: thomblake 08 June 2010 06:21:41PM 0 points [-]

Well the Monty Hall problem as stated never occurred on Let's Make a Deal. He's even on record saying he won't let a contestant switch doors after picking, in response to this problem.

Comment author: Douglas_Knight 09 June 2010 03:53:54AM *  0 points [-]

Robin's description is correct. I'm not sure what you're saying.

ETA: this thread has gotten ridiculous. I'm deleting the rest of my comments on it. The best source for info on Monty Hall is youtube. He does everything. One thing that makes it rather different is that it is usually not clear how many good and bad prizes there are.

Comment author: Jack 09 June 2010 12:09:57PM 4 points [-]

I'm really shocked by the reactions of the mathematicians. I remember solving that problem in like the third week of my Intro to Computer Science Class. And before that I had heard of it and thought through why it was worth switching. I didn't realize it caused so much confusion as recently as 20 years ago.

Comment author: JoshuaZ 09 June 2010 01:39:49PM 2 points [-]

The problem causes a lot of confusion. There are studies which show that this is in fact cross-cultural. It seems to deeply conflict with a lot of heuristics humans use for working out probability. See Donald Granberg, "Cross-Cultural Comparison of Responses to the Monty Hall Dilemma" Social Behavior and Personality, (1999), 27:4 p 431-448. There are other relevant references in Jason Rosenhouse's book "The Monty Hall Problem." The problem clashes with many common heuristics. It isn't that surprising that some mathematicians have had trouble with it. (Although I do think it is surprising that some of the mathematicians who have had trouble were people like Erdos who was unambiguously first-class)

Comment author: arundelo 09 June 2010 02:36:02PM *  3 points [-]


Wow! I looked this up and it turns out it's described in a book I read a long time ago, The Man Who Loved Only Numbers (do a "Search Inside This Book" for "Monty Hall"). Edit: In this book, the phrase "Book proof" refers to a maximally elegant proof, seen as being in "God's Book of Proofs".

I encountered the problem for the first time in a collection of vos Savant's Parade pieces. It was unintuitive of course, but most striking for me was the utter unconvincibility of some of the people who wrote to her.

Comment author: thomblake 09 June 2010 02:41:09PM 2 points [-]

the utter unconvincibility

Yes, my fallback if my intuition on a probability problem seems to fail me is always to code a quick simulation - so far, it's always taken on about a minute to code and run. That anyone bothered to write her a letter, even way back in the 70's, is mind-boggling.

Comment author: AlephNeil 09 June 2010 02:50:45PM 0 points [-]

Yeah it's remarkable isn't it?

I suppose the thing about the Monty-Hall problem which makes it 'difficult' is that there is another agent with more information than you, who gives you a systematically 'biased' account of their information. (There's an element of 'deceitfulness' in other words.)

An analogy: Suppose you had a coin which you knew was either 2/3 biased towards heads or 2/3 biased towards tails, and the bias is actually towards heads. Say there have been 100 coin tosses, and you don't know any of the outcomes but someone else ("Monty") knows them all. Then they can feed you 'biased information' by choosing a sample of the coin tosses in which most outcomes were tails. The analogous confusion would be to ignore this possibility and assume that Monty is 'honestly' telling you everything he knows.

Comment author: MartinB 10 June 2010 01:34:13AM 1 point [-]

Expert confidence. I read vos Savants book with all the letters she got and like how the problem seems to really be a test for the mental clarity and politeness of the actors involved.

Anyone knows of problems that get similarly violent reactions from experts?

Comment author: thomblake 09 June 2010 01:43:43PM 1 point [-]

From Wikipedia:

Monty Hall did open a wrong door to build excitement, but offered a known lesser prize—such as $100 cash—rather than a choice to switch doors. As Monty Hall wrote to Selvin: And if you ever get on my show, the rules hold fast for you—no trading boxes after the selection. (Hall 1975)

The citation is from a letter from Monty himself, available online here.

I'm not sure how the article you linked to is relevant. It does describe an instance of Monty Hall actually performing the experiment, but it was in his home, not on the show.

Comment deleted 09 June 2010 03:25:21PM [-]
Comment author: RobinZ 09 June 2010 04:16:59PM 1 point [-]

thomblake's remark was relevant too, though - from what I said, you might imagine that Monty Hall let people switch on the show. All the clarifications are relevant and good.

Comment author: RobinZ 09 June 2010 11:49:20AM 0 points [-]

Aaargh! And I had upvoted that, believing a random Internet comment over a reliable offline source! That's a little embarrassing.

The article is awesome, by the way. Thanks!

Comment author: thomblake 09 June 2010 01:44:18PM 1 point [-]

See response here