User talk:Martin Hogbin/Morgan Criticism

My criticism has now been moved to main user space to allow this page to be used for related discussion

Misquotation of the problem statement

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)

The question being asked

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)

But Morgan describe the unconditional situation as a slightly different problem. Martin Hogbin (talk) 22:55, 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)

They added some humour maybe. Martin Hogbin (talk) 22:55, 1 January 2010 (UTC)

The question being answered

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.

Yes, of course people should be aware of what they ask but, in the case of a non-expert member of the public asking a question in a popular magazine, it is fair to expect the respondent to make reasonable assumptions about the intent of the questioner. This is exactly what vos Savant did. Her only failing was to not make clear initially that she took the host's legal door choice to be random. Martin Hogbin (talk) 12:42, 10 January 2010 (UTC)

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)

Yes, I agree. Morgan misquote, then misinterpret the question to make it conditional. Martin Hogbin (talk) 12:42, 10 January 2010 (UTC)

More Importantly, What 'Question(s)' Are They Calling False?

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)

Glkanter, do you accept that Morgan have answered my stated problem correctly? Martin Hogbin (talk) 20:49, 10 January 2010 (UTC)

I think I've studiously avoided certain whole threads and topics on these discussion pages. Properly stating the MHP is not something I've been keen to battle since the old days of the '5 agreed bullet points', back in the spring (?).

Who cares, about Morgan anyways? And why? What new information did they bring to Selvin's or vos Savant's paradox? That the simple solutions are false? It's published. I don't believe it. Neither do the Professors who continue to teach it today. Nor do I think Rick's OR interpretations of Morgan have any more meaning today then they did in October, 2008, and certainly don't belong in the article. Thanks for asking, I have nothing to offer on this. Just waiting on Mediation, or whatever other ways there are to allow the will of the consensus to proceed with editing the article. Glkanter (talk) 21:45, 10 January 2010 (UTC)

The question that they have in fact answered

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.

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. The car and the goats were placed randomly. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. The host always opens another door and always shows a goat. He then says to you (the host always makes the offer to switch to the other unopened door), "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? More specifically, based only on the above information and given you've initially picked door No. 1 and then the host has opened door No. 3, what is the probability the car is behind door No. 1 and what is the probability the car is behind door No. 2?

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 find it hard to believe that you think that the authors of a paper to be published in a peer-reviewed journal about an extremely contentious topic can be excused from clearly describing the problem that they are addressing just because some of the vital information was stated by one of the authors that they attack. Do you really thing that the reader should be expected to somehow know that it is vos Savant's work that contains the missing information and then get a copy of this just so that the paper makes sense. The Morgan paper must stand in its own right, especially in the light of the ridiculous claim it makes to have found 'an elegant solution that assumes no additional information'.

Regarding the vos Savant scenario, Morgan do not even say something like, 'Vos Savant stated the basis on which she solved the problem, we shall now make the same assumptions that she did'. Had they said that they would still have been wise to have restated the assumptions on which they were working, but at least it would have made clear that there were some assumptions not stated in their paper.

What Morgan actually say is '...vos Savant makes it clear that the problem that she wishes to solve is one in which the host never has the option of showing the car. ... If these are the rules of the game, which we will call the vos Savant scenario...'. Morgan clearly mean the vS scenario to just refer to the fact that the host never shows the car. There is not a word about adopting her other, very reasonable assumptions.

Vos Savant made an excellent job of interpreting a question from a member of the public to a general interest magazine sensibly, with the possible exception that she failed to initially make clear that she took the host to act randomly when choosing a goat door to open if the player had initially chosen the car. There was therefore a case for a short letter, maybe to 'Parade', saying that vS had omitted to specify an important detail and that, if it was assumed that the host did not act randomly, her answer could be incorrect. There is no case for a vitriolic and bellicose paper which fails to identify many of its own essential assumptions.

Whether vS took the host to act randomly is not that important but it is important that anyone who attempts to solve the problem acts consistently. Either you take the producer's initial car placement and the host's legal door choice to be random (as vS later claimed that she had done) or you take them both to be non-random, in which case the problem cannot be solved.

It least we agree on the problem that Morgan were, in fact, solving let us see if Glkanter accepts it also; if so this may reduce some of the argument. Martin Hogbin (talk) 20:47, 10 January 2010 (UTC)

I find it hard to believe you find the problem they're addressing in their paper to be so mysterious, and disagree about your interpretation of the tone. "Vitriolic and bellicose"? That seems a more appropriate description for your continued attacks on this paper. Did they run over your dog or something? I really don't get it. -- Rick Block (talk) 21:36, 10 January 2010 (UTC)

I do not find the problem that they are addressing mysterious, with some effort, I have worked out exactly what it is. I would challenge most people to read the paper then say exactly what problem Morgan are addressing. Note that there are other changes that I have made in my question to make Morgan's answer correct. The questions that we should be asking are: Is the question they are asking the question that Whitaker wanted answered? Have they correctly interpreted Whitakers intent? (sound familiar?) Is it the MHP?

I freely admit that I do not like the Morgan paper. Firstly because it turns a simple mathematical puzzle that most people get wrong into a complicated problem that most people cannot understand. It has been the cause of most of the argument here and it is the principle reason that the article has failed to address the MHP for the average reader.

Secondly their unprovoked attack on vos Savant is unbecoming of a paper published in a peer-reviewed journal. They start by calling her solution false, they include things like, 'Savant ... makes clear that the problem she wishes to solve', they use the term, 'vos Savant scenario', is if it were some bizarre scenario (yet then base their own solution tacitly on it) and then end the paper by saying '...confused the world's most intelligent person'. I am not the only person not to like the paper, Rosenhouse describes it as a 'bellicose and condescending essay'. Martin Hogbin (talk) 23:09, 10 January 2010 (UTC)

I think the Morgan et al paper is pedantic and dull, boastful and boring, misconceived and misleading. Such a pity that they turned a golden paradox into heap of rusty iron. Gill110951 (talk) 22:02, 17 January 2010 (UTC)

I agree. The paper is not well liked by many. Martin Hogbin (talk) 23:20, 17 January 2010 (UTC)

What is the Morgan 'conjuring trick'?

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.

Setting the scene

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.

The trick

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.

(if the car always is behind the chosen door the chance of winning by switching will always be 0). If the car had been placed at random however, the chance by switching is always 1/2 if the host opens his "strictly preferred door", and if he opens his "strictly avoided door" the chance by switching will always be 1. (That makes 2/3 in average). -- Gerhardvalentin (talk) 01:13, 5 March 2010 (UTC)

I am not sure what your point is here. If the car is never placed behind door 1 the probability of winning by switching is 1. The (Whitaker's) problem statement does not tell us how the car is placed thus, strictly speaking, the answer is indeterminate. See my question below. Martin Hogbin (talk) 11:30, 5 March 2010 (UTC)

Please pardon me, and please delete all of my stuff, I just wanted to underline that you are right: Some presumptions show to be really ridiculous. If the car "always" is placed behind the door selected then the chance by switching of course will always be zero. Ehh? - ridiculous presumption, like Morgan; he knows: The chance of picking the winning door is exactly 1/3, so the chance of winning by switching is always 2/3, and Morgans is just wisecracking, he says:
Anytime the host SHOWS that the car ACTUALLY is behind his preferred door, i.e. he cannot open his preferred door and he HAS exceptionally to open "his avoided door" then everyone knows that IN THIS CASE the chance by switching ACTUALLY is exactly 3/3. Ridiculous.
And every time (3/3) the host ACTUALLY opens "his" door, the car being behind the "other" unselected door (in 1/3 chance for switching=100%) or if he has two goats to chose from (in 2/3 chance for switching=0%) then, in average, the door selected IS SHOWN to have ACTUALLY a chance of 50%, and the chance by switching IS SHOWN to be ACTUALLY 50 % likewise.
But: That does not change the overall probability that remains unchanged: Overall chance by switching remains 2/3, so it is always wise to switch. Morgan cannot change this overall chance, the host just SHOWS in every case whether the car ACTUALLY IS NOT behind the door selected, i.e. the car ACTUALLY IS behind the door offered as an alternative, or whether it is behind the door selected with a probability of just 50 %, i.e. it is also behind the door offered as an alternative with a probability of remarkable 50 % likewise.
Ridiculous. And everyone applauds him, everyone bows to him. Please Martin delete all of this. Thank you. -- Gerhardvalentin (talk) 12:39, 5 March 2010 (UTC)

Gerhard please feel free to delete any of you comments if you wish but there is no need to do so. We are all learning here. many good mathematicians got the basic MHP wrong. In order to properly understand why the Morgan paper is wrong you first need to understand why it is right. This may seem odd but you have to understand how modern probability theory works. Please give me your answer to my question below and I will explain. Martin Hogbin (talk) 13:17, 5 March 2010 (UTC)

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 question

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)

If no information was revealed regarding the ACTUAL placement of the car, then the chance to choose the car is 1/3, they selected door #1, so the probability they selected the car in this special case also is 1/3. And - before the host opens a door - it is known that the pair of the two unselected doors has a chance of 2/3. Okay? -- Gerhardvalentin (talk) 13:33, 5 March 2010 (UTC)

OK, I was not meaning this to be the start of a Monty Hall show, just a very simple show where a player has to pick one of three doors.

Suppose now we have this game. You to think of a number from 1 to 3 and I have to guess what it is. What is the probability that I get it right? Martin Hogbin (talk) 13:44, 5 March 2010 (UTC)

Your chance to get my number is 1/3, I suppose. -- Gerhardvalentin (talk) 14:21, 5 March 2010 (UTC)

But in reality that is not correct. People do not choose numbers strictly randomly, so someone who knows which numbers are most likely to be chosen could do better than 1/3. Try this trick on your friends. Deal out four cards, face up in a row, from the top of a shuffled pack. Ask them to think of one without telling you. Now tell them that they have picked the second card from the left. This is correct surprisingly often. It is just what people do.

So, in modern probability theory, the 3 door question above would be represented by having a sample set with three events in it representing the car being behind doors 1, 2, and 3. In this very simple game, in which the player picks a door and they win whatever is behind it, the probability of winning the car is simply the probability that it has been placed behind door 1.

Now you may, as the person answering the question, choose to assume that the car has been placed uniformly at random, or you might say that as we have no information on how the car was placed we should apply the principle of indifference and take it to be uniformly distributed. The point is that, if you are not given the distribution of the car behind the three doors in the problem statement, and you do not make some stated assumptions in your answer (such as it is uniformly distributed because that is what they do on a game show) then the distribution has to be assumed to be arbitrary, in other words it could be absolutely anything. Martin Hogbin (talk) 14:56, 5 March 2010 (UTC)

Gerhard, have you lost interest? Martin Hogbin (talk) 23:15, 6 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)

Martin, it's difficult. Difficult because you have to consider various aspects: If you do KNOW that the car originally had been placed behind one of those three doors uniformly at random, the answer will be "1/3", then. But in case you just suppose that the car originally ... could have been placed uniformly at random, but having no information on how it really was placed, you should consider that there could have been some certain "bias" (of unknown extent) to put it behind one "special" door only. So, even if this is hypothetic only, and no one knowing exactly about, you would have to consider such "bias", even if you have no information about its degree / extent. Because the original question did not respond to that point. So you should provide for such a bias and allow for it.

But the original question responds to a second point, and it responds very clearly:

Please read the original question in the letter put to 'Parade': Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, (and I am adding: The host does not open all three doors altogether!

Read: "opens another door", i.e. ONE different door only) say No. 3, which has a goat. He then says to you "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Repeat: "ONE" different door will be opened, revealing a goat, just "another door". Not disclosing the contents of all three doors altogether, as Morgan suggests:

For in 1/3 of all cases a "strongly biased Morgan-Host" will unavoidably do so. This is my opinion on the "relevance" of Morgan's paper, just giving away "additional" information (Falk) on the actual result/constellation. As the "strongly biased host" never is provided for in the original question, one has to say that, as a matter of fact, the original question evidently implies an unbiased host. If the host can choose between two goats, he will always show ONE of them uniformly at random, as an evidence already being implied in the original question. Morgan, who made his own new rules for his own new game, has long ago been disqualified. Morgan has been beaten before he started his paper. I suppose he still doesn't know. May be, neither some others. --Gerhardvalentin (talk) 16:54, 19 July 2010 (UTC)

Gerhard, you seem to have muddled up your questions. The one you quote above is not that from Whitaker's letter to 'Parade' but an unconditional problem statement, about which there is no argument. Martin Hogbin (talk) 17:09, 19 July 2010 (UTC)

Just saw that, and you're right. Gerhardvalentin (talk) 17:29, 19 July 2010 (UTC) - now. Gerhardvalentin (talk) 18:00, 19 July 2010 (UTC)