What you "cannot" do is assign probabilities to the cases. You can still - and my point is "must" - treat them as random variables, which have unknown probability distributions.Except that you cannot, and you know that you cannot. — Srap Tasmaner
Define "interact."This makes no sense to me. Initial distribution of what? If these are pairs of envelopes from which will be chosen the pair that the player confronts, then not only is this sample space unknown to the player, she never interacts with it. She will face the pair chosen and no other. — Srap Tasmaner
Yes, it is tempting to say that if the game is only being played once then the shape of the initial distribution(*) isn't relevant to the definition of the player's known problem space. That's a fair point, and I may not have been sufficiently sensitive to it. — Pierre-Normand
No, it isn't fair to say that. No more than saying that the probability of heads is different for a single flip of a random coin, than for the flips of 100 random coins — JeffJo
The probabilist reasons from a known population to the outcome of a single experiment, the sample. In contrast, the statistician utilizes the theory of probability to calculate the probability of an observed sample and to infer from this the characteristics of an unknown population.
Mathematical Statistics with Applications, Wackerly, Mendenhall, Scheaffer
What is the source of the paradox in your view? — Andrew M
(...) This is the fallacy. You reason from the fact that, given the criterion of success, you would have a 1 in 2 chance of picking the envelope that meets that criterion, to a 1 to 2 chance that the unknown criterion of success is the one your chosen envelope meets. (...) — Srap Tasmaner
I suggested at one point in this thread that if told the value of the other envelope instead of your own, then you would want not to switch; I found this conclusion absurd but my interlocutor did not. Go figure. — Srap Tasmaner
But in the ideal case — Pierre-Normand
I know we are reaching an equivalent conclusion. My point is that the framework that it fits into may be different. These concepts can seem ambiguous to many, which is the fuel Bayesians, Frequentists, Subjectivists, Objectivists, Statisticians, and Probablists use to denigrate each other through misrepresentation.Suppose, again, that a die throwing game will be played only once (i.e., there will be only one throw) with a die chosen at random between two oppositely biased dice as earlier described. ... — Pierre-Normand
I believe there is not a paradox here but a fallacy. — Srap Tasmaner
Claiming this case is "ideal" is an entirely subjective standard pumped full of observational bias. — Jeremiah
(...) That distribution is an unknown function F1(x). After picking high/low with 50:50 probability, the value in our envelope is a new random variable V. Its distribution is another unknown function F2(v), but we do know something about it. Probability theory tells us that F2(v) = [F1(v)+F1(2v)]/2. But it also tells us that the distribution of the "other" envelope, random variable Y, is F3(y) = [F1(y)+F1(2y)]/2. Y is, of course, not independent of V. The point is that it isn't F3(v/2)=F3(v)=1/2, either. — JeffJo
Looking in the envelope does change our role from that of the game player, to the gamemaster. Just like seeing the color of your die does not. Simply "knowing" v (and I use quotes because "treat it as an unknown" really means "treat it as if you know the value is v, where v can be any *single* value in the range of V") does not change increase our knowledge in any way.
A random variable is defined by a real world function — Jeremiah
"I don't know" is not the same as "There is a 50% chance of each." — Srap Tasmaner
A normal prior would actually make more sense, as empirical investigations have shown it robust against possible skewness. — Jeremiah
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