A while ago I posted a question about this counter-intuitive probability paradox: If you have a lion behind 1/3 doors, and you open a door and find it's not there, then you increase your chances by switching your original guess. This doesn't make sense to me, but I would have to stare at it a while to make sense of it. Assuming it's true -
Wouldn't it effect a guessing strategy in actual situations? Suppose you think that cancer is caused by genetics, carcinogens, or poor diet. You fsvor carcinogens, and all guesses are unknown probability. Then you find more evidence against genetics. Now even though the evidence rules out genetics, does it mean you should switch to poor diet? Of course this is messier than the hypothetical Monty Hall problem, but you can derivatize it by calling it a heuristic for further research, rather than a final guess. So what's key is your ignorance is equal between the three possibilities.
Or doesn't Monty Hall have legs of this kind? I note that the wikipedia describes "quantum mechanical game theory."
Any thoughts welcome.
jb
"When you come to a fork in the road, take it" - Yogi Berra.
What you are describing sounds like a Bayes network:
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Where, as in your example, in a certain bayes network modeling cancer causes, evidence against genetics would increase the conditional probability of poor diet, etc.
Is how the Monty Hall problem would be expressed in a Bayes network, where knowledge of the occurrence of C alters the conditional probabilities of the upstream events
This "paradox", as I've heard it described before, has some additional details which may make things clearer.
It's usually specified as being a game show. You choose a door (one chance in three of it having a prize behind it). After you make your choice, the game-show host opens one of the other doors and shows you that the big prize was not behind it. You then have the option of choosing to switch your initial choice.
This scenario is different than the simpler one you suggested... because there's a game-show host, who has knowledge that you lack (where the prize is) and acted *selectively* on the basis of that knowledge.
When you make your initial choice, there are two possibilities: you picked the door with the prize (1/3 chance) or did not (2/3 chance).
If you picked the door with the prize (1/3 of the time), then the host can pick either of the other doors at random and show it to you - neither has the prize and it doesn't matter which one he shows you. In this case (1/3 of the time) changing your original decision is bad, as you lose the prize.
If you had originally picked a door without the prize (2/3 of the time), then things are different. The host knows which of the other doors has the prize... and the host *must* pick the other door (which has no prize) to open for you. In this situation (2/3 of the time) changing your initial decision is good, as it wins you the prize.
So... in this Monty Hall scenario, you get the best probability by choosing one door at random, letting Monty show you an empty door, and switching your choise to the third door. You end up with a 2/3 chance of winning the big prize.
Now, if it's a tiger behind the door, and you don't want to be eaten, your best bet is to choose at random, let Monty show you an empty door you aren't allowed to escape through :-) and then stick with your original choice. 2/3 of the time it will not have the tiger.
I don't think the same principle applies here. You don't have the same "rules" to work with. That is, you don't have a guaranteed gold-plated assurance that cancer is definitely the result of precisely one of these three things, and you don't have an omniscient "oracle" such as Monty Hall who can declare one of the choices as "definitely not the case".
Cancer could be caused by two or more of these things, in combination, or by a fourth or fifth factor which you do not know. That complexity is absent in the simple, clearly-ruled Monty Hall problem.
I certainly will have to look at it more closely. The many formulations I am hearing here take the paradox away - The formulation is so counter-intuitive that many world-class mathematicians said it couldn't be true. (Erdos, plus a poll taken at an academic math conference.)
I will have to do my homework o this, but you must ask yourself, does my explanation account for the huh? response from many professional mathematicians, with that huh? state lasting weeks in some cases? (Erdos)
jb
"We have the world's best school system in DC, if you don't count detail that
83% of kids can't read at graduation." - Mayor Berry, DC.
** It makes more obvious sense if you increase the number of doors to say
100. You pick one with a chance of being right only 1%.
The MC then opens the 98 doors that are empty.
Changing your pick increases your chance to a near certainty.
** Just a little clear thinking would save time.
It is a very simple problem in probability theory, ie high school math.
If you stick with your original pick, making no use of the information given by the open doors, the chance you are wrong is 99%. If you had both picks, you would be right 100%.
So changing your pick (the only alternative) gives a 99% chance of being right.
What you are looking for is Bayesian inference named after the Reverand Bayes who first formulated it and first used by Laplace to weight Saturn when he called it the "Principle of insufficient reason".
The work of Jeffries in the first half and Ed Jaynes in the latter half of the previous century were seminal in getting it established on a firm mathematical foundation. It allows you to answer questions like a 6 sided die sides numbered 1..6 gives a long term average of 4.5. Estimate the probability of throwing a six on the next go. This is a pretty good introduction although now a bit dated:
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These days Bayesian analysis is more or less routine when you want to extract the very last trace of hard to obtain signal out of noise.
It is also intimately related to decision theory and game theory (and high frequency trading where the suits thought they could make the risk vanish by using it). Problem is you get can killed by something very unusual that they have forgotten to include in their modelling.
Some high frequency share dealing tricks *are* risk free like given a large client buy order buying the stuff on your own account with a much faster system and then selling on it to the client at a markup 2ms later. Apparently this doesn't count as insider trading!
If that is true then the US public school system is horribly broken.
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