My criticism has now been moved to main user space to allow this page to be used for related discussion
It seems indeed that they misquoted, given the other quotes. Which is in their benefit of not having to use any assumptions offering the "elegant" solution. Heptalogos ( talk) 22:23, 1 January 2010 (UTC)
I do not agree. There are two ways to solve this adequately: interpret the question literally, or interpret as reasonable. Actually they do both, which is quite good. Their reasonable interpretation is what they call the unconditional situation. Nevertheless, the exact interpretation is the best from an objebtive, and surely from a scientific point of view. I don't see the problem. Heptalogos ( talk) 22:22, 1 January 2010 (UTC)
Yes, only because it's unconditional. This could theoretically make any difference. We know this is beta-crap, but they stick to the rules, as they should, as "reproducing" scientists. We have actually found a gap in those rules, or even absence. Heptalogos ( talk) 23:47, 1 January 2010 (UTC)
Their objective was essentially not to provide an answer to the questioner in Parade -as good statisticians should execute their assignment- but to provide wisdom to the discussions between many. Probably they wouldn't even have done that without the extra challenge of providing the ultimate elegant solution. Let's assume they are right about the conditional issue (which they probably believe honoustly). In that case it seems perfectly justified to mention the exact aspects of it. Again, they did not just criticize; they added a lot of value. Heptalogos ( talk) 22:45, 1 January 2010 (UTC)
The assumptions about randomness made by Morgan are reasonable. Actually they approved these assumptions already made by vos Savant, and use the same.
Why do you introduce a new issue, like the known history of the game? Should be made explicit also the assumption that the world still exists after opening a door? However, a known host strategy is assumed, from which the past is not excluded.
It is quite clear that in Morgan's solution, the door numbers are as given: only 1 and 3. This is correct when solving the exact question, as they do. If in the question asked other numbers were used, the solution would be exactly the same, except for the numbers.
The host, who knows what's behind them. This phrase is very irrelevant; we really don't know what's the use of it. It may seem reasonable that it assumes an intention, but what precise intention? People (Whitaker) should really be aware of what they exactly ask and how they formulate the question. Since the issue is interpreted literally, there is no information in it.
The specific door numbers are indeed what makes the problem conditional. Actually they are used as examples in the question being answered too, because it could be any number, though specific. Heptalogos ( talk) 23:41, 1 January 2010 (UTC)
I dunno. Because it isn't the one that begins, 'Suppose you're on a game show...' Vos Savant, herself, points this out in her letter to American Statistician.
What else is new in their paper? Selvin did conditional solutions back in 1975. Glkanter ( talk) 15:42, 10 January 2010 (UTC)
The question Morgan et al. address is not some bizarre variant they concocted on their own, but rather their interpretation of the original problem from vos Savant's column plus her clarifications from her subsequent columns, which is obviously why they call it the "vos Savant scenario". Rather than rewrite the problem statement completely, another way to present the problem they address is to annotate Whitaker's original problem statement (as published by vos Savant). For example, I think the following is another way to specify the problem Morgan et al. address.
With the possible exception of whether vos Savant meant the host must choose randomly between two goats, I don't think there's any significant argument about this up to "More specifically". Whether the host must choose randomly in this case is really not the point, but it does allow the main point to be more easily seen. I think the main point of contention is the sentence starting "More specifically". -- Rick Block ( talk) 19:38, 10 January 2010 (UTC)
I have often said that Morgan et al perform a conjuring trick. Just like the conjurer's rabbit hat he pulls out of a hat is a real rabbit, Morgan's answer is a real answer, but all is not quite what it seems.
First it is important to consider the paradigm that Morgan use in their paper. Rather than the more traditional Bayesian or frequentist, ideas Morgan use the more modern description of probability theory. This is a formal mathematical system based on elements in a sample space. The important point here is that, as the article says, 'The modern definition does not try to answer how probability mass functions are obtained; instead it builds a theory that assumes their existence'. In other words the somewhat philosophical and really important questions, such as, 'from what state of knowledge are we addressing this question?' are included in the sample space at the start. This is the conjurer's black cloth that he drapes over the table so that you cannot see what is happening underneath it.
I may not have used the correct terminology above as I am no expert on the subject. The point that I am making is that certain assumptions can made right at the start of the problem and presented as a fait accompli. This is how the trick is done.
Like many good conjuring tricks the clever bit is done before the action appears to start.
In Whitaker's question, which is what Morgan claim to be addressing, there are three undefined distributions: the distribution with which the producer originally places the car behind the door; the distribution with which the player chooses a door; and the distribution with which the host chooses a legal (one that reveals a goat) door to open. None of these distributions is given in the original question. The player's initial door choice turns out not to be that important in most cases.
So Morgan start with a sample space in which the probability that the car is behind any given door is 1/3. That seems reasonable but in fact, by doing this, they have already placed the rabbit in the hat, all that needs to be done is to pull it out and amaze the audience.
What should they do? I am not sure there is any absolutely correct answer but any good mathematician would at least expect a consistent treatment. As none of the distributions is known we could apply the principle of indifference and take them all to be uniform. This is what Selvin did right at the start (after his second letter). The answer then is the classic MHP in which the probability of winning by switching is 2/3.
The alternative, and arguable more strictly correct, approach is to say that we do not know any of these distributions and thus must assume that they could be anything (the producer might always put the car behind door 1 or the host might always pick the highest numbered legal door for example). The problem with this approach is that it gives a very simple and boring answer. The probability of winning by switching can be anything from 0 to 1.
What Morgan do is to say (in my words), 'we are proper statisticians, thus we must take the host's choice to be undefined, but we can represent it with a parameter q', now, having already tacitly and inconsistently assumed the initial distribution of the car to be uniform, they do a simple mathematical calculation and pull an 'elegant solution' out of the hat. Easy when you know how. Martin Hogbin ( talk) 19:16, 4 March 2010 (UTC)
A car in a game show is placed behind one of three doors (numbered 1-3). A player chooses door 1, what is the probability that they choose the car? Martin Hogbin ( talk) 11:30, 5 March 2010 (UTC)
NO, but actually lack of time. Will soon be back with some questions. Regards, Gerhardvalentin ( talk) 01:36, 7 March 2010 (UTC)
My criticism has now been moved to main user space to allow this page to be used for related discussion
It seems indeed that they misquoted, given the other quotes. Which is in their benefit of not having to use any assumptions offering the "elegant" solution. Heptalogos ( talk) 22:23, 1 January 2010 (UTC)
I do not agree. There are two ways to solve this adequately: interpret the question literally, or interpret as reasonable. Actually they do both, which is quite good. Their reasonable interpretation is what they call the unconditional situation. Nevertheless, the exact interpretation is the best from an objebtive, and surely from a scientific point of view. I don't see the problem. Heptalogos ( talk) 22:22, 1 January 2010 (UTC)
Yes, only because it's unconditional. This could theoretically make any difference. We know this is beta-crap, but they stick to the rules, as they should, as "reproducing" scientists. We have actually found a gap in those rules, or even absence. Heptalogos ( talk) 23:47, 1 January 2010 (UTC)
Their objective was essentially not to provide an answer to the questioner in Parade -as good statisticians should execute their assignment- but to provide wisdom to the discussions between many. Probably they wouldn't even have done that without the extra challenge of providing the ultimate elegant solution. Let's assume they are right about the conditional issue (which they probably believe honoustly). In that case it seems perfectly justified to mention the exact aspects of it. Again, they did not just criticize; they added a lot of value. Heptalogos ( talk) 22:45, 1 January 2010 (UTC)
The assumptions about randomness made by Morgan are reasonable. Actually they approved these assumptions already made by vos Savant, and use the same.
Why do you introduce a new issue, like the known history of the game? Should be made explicit also the assumption that the world still exists after opening a door? However, a known host strategy is assumed, from which the past is not excluded.
It is quite clear that in Morgan's solution, the door numbers are as given: only 1 and 3. This is correct when solving the exact question, as they do. If in the question asked other numbers were used, the solution would be exactly the same, except for the numbers.
The host, who knows what's behind them. This phrase is very irrelevant; we really don't know what's the use of it. It may seem reasonable that it assumes an intention, but what precise intention? People (Whitaker) should really be aware of what they exactly ask and how they formulate the question. Since the issue is interpreted literally, there is no information in it.
The specific door numbers are indeed what makes the problem conditional. Actually they are used as examples in the question being answered too, because it could be any number, though specific. Heptalogos ( talk) 23:41, 1 January 2010 (UTC)
I dunno. Because it isn't the one that begins, 'Suppose you're on a game show...' Vos Savant, herself, points this out in her letter to American Statistician.
What else is new in their paper? Selvin did conditional solutions back in 1975. Glkanter ( talk) 15:42, 10 January 2010 (UTC)
The question Morgan et al. address is not some bizarre variant they concocted on their own, but rather their interpretation of the original problem from vos Savant's column plus her clarifications from her subsequent columns, which is obviously why they call it the "vos Savant scenario". Rather than rewrite the problem statement completely, another way to present the problem they address is to annotate Whitaker's original problem statement (as published by vos Savant). For example, I think the following is another way to specify the problem Morgan et al. address.
With the possible exception of whether vos Savant meant the host must choose randomly between two goats, I don't think there's any significant argument about this up to "More specifically". Whether the host must choose randomly in this case is really not the point, but it does allow the main point to be more easily seen. I think the main point of contention is the sentence starting "More specifically". -- Rick Block ( talk) 19:38, 10 January 2010 (UTC)
I have often said that Morgan et al perform a conjuring trick. Just like the conjurer's rabbit hat he pulls out of a hat is a real rabbit, Morgan's answer is a real answer, but all is not quite what it seems.
First it is important to consider the paradigm that Morgan use in their paper. Rather than the more traditional Bayesian or frequentist, ideas Morgan use the more modern description of probability theory. This is a formal mathematical system based on elements in a sample space. The important point here is that, as the article says, 'The modern definition does not try to answer how probability mass functions are obtained; instead it builds a theory that assumes their existence'. In other words the somewhat philosophical and really important questions, such as, 'from what state of knowledge are we addressing this question?' are included in the sample space at the start. This is the conjurer's black cloth that he drapes over the table so that you cannot see what is happening underneath it.
I may not have used the correct terminology above as I am no expert on the subject. The point that I am making is that certain assumptions can made right at the start of the problem and presented as a fait accompli. This is how the trick is done.
Like many good conjuring tricks the clever bit is done before the action appears to start.
In Whitaker's question, which is what Morgan claim to be addressing, there are three undefined distributions: the distribution with which the producer originally places the car behind the door; the distribution with which the player chooses a door; and the distribution with which the host chooses a legal (one that reveals a goat) door to open. None of these distributions is given in the original question. The player's initial door choice turns out not to be that important in most cases.
So Morgan start with a sample space in which the probability that the car is behind any given door is 1/3. That seems reasonable but in fact, by doing this, they have already placed the rabbit in the hat, all that needs to be done is to pull it out and amaze the audience.
What should they do? I am not sure there is any absolutely correct answer but any good mathematician would at least expect a consistent treatment. As none of the distributions is known we could apply the principle of indifference and take them all to be uniform. This is what Selvin did right at the start (after his second letter). The answer then is the classic MHP in which the probability of winning by switching is 2/3.
The alternative, and arguable more strictly correct, approach is to say that we do not know any of these distributions and thus must assume that they could be anything (the producer might always put the car behind door 1 or the host might always pick the highest numbered legal door for example). The problem with this approach is that it gives a very simple and boring answer. The probability of winning by switching can be anything from 0 to 1.
What Morgan do is to say (in my words), 'we are proper statisticians, thus we must take the host's choice to be undefined, but we can represent it with a parameter q', now, having already tacitly and inconsistently assumed the initial distribution of the car to be uniform, they do a simple mathematical calculation and pull an 'elegant solution' out of the hat. Easy when you know how. Martin Hogbin ( talk) 19:16, 4 March 2010 (UTC)
A car in a game show is placed behind one of three doors (numbered 1-3). A player chooses door 1, what is the probability that they choose the car? Martin Hogbin ( talk) 11:30, 5 March 2010 (UTC)
NO, but actually lack of time. Will soon be back with some questions. Regards, Gerhardvalentin ( talk) 01:36, 7 March 2010 (UTC)