Expectation of how many winners

[caveman1917]

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".

 

[Ken G]

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.

 

[caveman1917]

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?

 

[mfb]

This went from a discussion of probabilities to a discussion about the discussion style, without hope of an agreement. I closed the thread.