Expectation of how many winners

  • Thread starter Ken G
  • Start date

As a winner notified this way, how do you update <N>?

  • You expect that N is probably fairly large

    Votes: 0 0.0%
  • You still don't know how large to expect N, but larger than before

    Votes: 0 0.0%
  • You make no changes to your expectations on N

    Votes: 5 100.0%

  • Total voters
    5
  • Poll closed .
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But in your OP here you literally state that you have "no meaningful expectation on N", yet when you are notified in secret your expectation goes up. How do you explain that?
It is no problem to discuss update factors even if we did not decide on an initial probabilty distribution.

Then how do you explain that it doesn't change when I would make the secret information that I am a winner public?
It would change, if you assume that all winners would post that on twitter in the same way, and if you are the first one to post it.

I would hope that to be true.
It is true.
 
It is no problem to discuss update factors even if we did not decide on an initial probabilty distribution.
Why would it be, and how is that relevant to the claims being made? Do you agree with Ken that "the expectation doesn't go up because I can not choose a meaningful prior probability and therefor there is nothing to update" is a correct answer?

It would change, if you assume that all winners would post that on twitter in the same way, and if you are the first one to post it.
Which again has nothing to do with the information being "private" or "public", but everything to do with your information set containing either "I am a winner" or "I am the first winner". You can easily see this by slightly changing your example, where instead of posting it on twitter everybody is supposed to post it on a physical messageboard in some building. Suppose you, as a winner, go to that building and when you want to enter the security guard privately whispers in your ear: "You are the first one to come post it". Would you now not update on that information because it isn't public yet? Would you wait updating until you actually get to the messageboard and have posted your result publicly on it, even if you have no reason to think that the security guard is lying?

It makes no difference whatsoever whether the information that you have is private or public (it's not even clear how you would define those concepts anyway), the only reason you're getting that result is because your examples are stated in such a way that the information of "being a winner" is given privately and "being the first winner" is given publicly.

The private/public thing is a huge red herring. As are the "feelings of, among others, specialness" that are being presented as a basis for getting the answer.

It is true.
Is there a requirement to use mainstream mathematics on the forum?
 
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Why would it be, and how is that relevant to the claims being made?
Relative probabilities are independent of the priors.
Do you agree with Ken that "the expectation doesn't go up because I can not choose a meaningful prior probability and therefor there is nothing to update" is a correct answer?
I do not agree, and I do not see where such a claim has been made.

It makes no difference whatsoever whether the information that you have is private or public (it's not even clear how you would define those concepts anyway), the only reason you're getting that result is because your examples are stated in such a way that the information of "being a winner" is given privately and "being the first winner" is given publicly.
Right.

Is there a requirement to use mainstream mathematics on the forum?
Sure (if you use mathematics).
 
Relative probabilities are independent of the priors.
Not exactly. Relative posterior probabilities depend on the relative prior probabilities. What you probably mean is that the factor with which the relative prior probability is updated to form the relative posterior probability is independent of the relative prior probability. You can see this by writing out Bayes' theorem for any two hypotheses and dividing the equations.

I do not agree, and I do not see where such a claim has been made.
->
No one here cares about a thread on some other forum. If they are wondering, the simple answer to your question is that what was resolved in that thread is that any extent to which you increase your expectation on N depends on what your expectation was going in, so if you had no expectation going in, there is nothing to update and no change in an expectation that doesn't exist.
The "having no meaningful expectation on N" answer is, in Ken's language, the statement above. That is, he claims, because a flat probability distribution over N has a "meaningful" expected value.

Sure (if you use mathematics).
What if you use vague philosophy that contradicts mainstream mathematics?
 
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Not exactly. Relative posterior probabilities depend on the relative prior probabilities. What you probably mean is that the factor with which the relative prior probability is updated to form the relative posterior probability is independent of the relative prior probability. You can see this by writing out Bayes' theorem for any two hypotheses and dividing the equations.
That's what I mean, and that's what has been discussed all the time.

The "having no meaningful expectation on N" answer is, in Ken's language, the statement above. That is, he claims, because a flat probability distribution over N has a "meaningful" expected value.
Well, you cannot calculate an expectation value without a prior distribution, but you can correctly state that the expectation value will increase/decrease/whatever (depends on the setup) for all non-trivial prior distributions.

What if you use vague philosophy that contradicts mainstream mathematics?
Can we stop that meta-discussion?
 
That's what I mean, and that's what has been discussed all the time.
Strictly speaking that is what should have been discussed all the time, but it wasn't [ETA: it was, but not all the time].

Well, you cannot calculate an expectation value without a prior distribution, but you can correctly state that the expectation value will increase/decrease/whatever (depends on the setup) for all non-trivial prior distributions.
Yes exactly, it's simply Bayes' theorem, it's not particularly difficult. For the question "unknown N chosen out of 1 million, you are a winner, do you update for larger N?" the answer is "yes for all possible priors other than those with a 100% spike on some value". It is not "no because having a prior distribution means you have meaningful information on N because you can calculate an expected value, and if you have no prior distribution then there's nothing to update".
 
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Ken G

Gold Member
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Everyone understands Bayes theorem on this forum, including me. All you are saying is you don't understand what I am saying, which I already know. But since mfb might be wondering if your characterization of the situation is fair, I will summarize what I am invoking this puzzle to point out:

1) Details about how you gain your information, and what assumptions you make before you even get that information, have a significant impact on the conclusions you draw, so what you get out is a simple, yet surprisingly sensitive, function of what you put in.

2) If you have a prior expectation on N, then information you garner can increase that expectation. For example, if you are told you are in a generic class of winners, then your expectation on N increases by the factor <N2>/<N>2, which may not have been derived yet but it is straightforward-- but since it depends on your initial expectation <N>, you need to have an initial expectation or that increase factor is meaningless.

3) If you do not have a prior expectation on N, then it is incorrect to claim your expectation is that there is a flat probability distribution that applies to N. That is simply incorrect logic, it is just like saying "everything that I know nothing about has a 50% chance of happening, because it either will happen, or it won't happen." That's a flat probability distribution too, but the logic behind it is fruitless, and has no place in any real probability discussion. This is simply because any probability distribution depends on how you count the equally-likely elements that make up that distribution, and often this is impossible to do without significant prior information. Obviously, I can get flat distributions over many different choices of variable, and they will not even be consistent with each other, let alone with reality.

4) There is never any situation where you get a different expectation on N just because you are you-- it is always about the information you have, such that the instant you share all your information with everyone else, they must have the same expectation on N that you do. In particular, there is never any situation where you could be in a position of knowing something that you "just can't convince anyone else of because they are not you." That's always false logic to conclude that, yet we do see that logic in many situations, such as homeopathic remedies, astrological forecasts, and claims that quantum suicide can be tested by an individual but not by a scientific establishment. Those claims are all equally bad logic, and no one on this thread has argued them, so I won't bother to mention who has argued them elsewhere because that's extraneous to this thread. If that discussion comes up, it should be in a different thread, but since the conclusion is quite clear, there's not even a need for it.
 
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Everyone understands Bayes theorem on this forum, including me.
You meant "excluding you".

All you are saying is you don't understand what I am saying, which I already know.
Then present some learning materials.

But since mfb might be wondering if your characterization of the situation is fair, I will summarize what I am invoking this puzzle to point out:
Here's an easier way to find out: Consider this question:
Imagine you are 1 of a million people who put their names in a hat. A number N of names is chosen from the hat, but you have no idea what N is, except that N > 0. You are informed that your name has been selected! Is it true that you can conclude with good probability reasoning that, most likely, N is fairly large?
(where we define "fairly large" as "larger than it was")

and consider these possible answers:
"Yes that is valid reasoning"
"No that is a logical fallacy"
What is the answer? Please answer yes or no and then provide a rigorous argument for your answer.

That's always false logic to conclude that, yet we do see that logic in many situations, such as homeopathic remedies, astrological forecasts, and claims that quantum suicide can be tested by an individual but not by a scientific establishment. Those claims are all equally bad logic, and no one on this thread has argued them, so I won't bother to mention who has argued them elsewhere because that's extraneous to this thread. If that discussion comes up, it should be in a different thread, but since the conclusion is quite clear, there's not even a need for it.
I disagree. Please provide evidence of someone arguing each one of those claims (homeopathic remedies, astrological forecasts, QS as valid reasoning) on another forum. We wouldn't want to think you're just making a few of those up to poison the well, would we?
 
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This went from a discussion of probabilities to a discussion about the discussion style, without hope of an agreement. I closed the thread.
 

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