4 ms·

The problem is that "uniform distribution" is hard to define. An even distribution of water/wine values is not the same as an even distribution of wine/water v

by bhk 8y ago

The problem is that "uniform distribution" is hard to define. An even distribution of water/wine values is not the same as an even distribution of wine/water values. Thinking in terms of possible states, you can choose an even distribution of x from 0 to 1 and define: water/wine = x/(1-x) wine/water = (1-x)/x (which avoids being water-biased or wine-biased)

sjg007 8y ago
So this is defining the random variable not as the ratio but as the implicit function that computes the ratio? So we are really looking at the density function of the hidden variable X? For the original problem it feels like the interval constraints on the ratio are causing the issue since they will be inverse. But maybe I am not clear on that...
- bhk 8y ago
  The problem is not completely specified. However, in the study of probability and statistics there is a tendency to charge ahead and offer answers to incompletely specified problems, using e.g. the Principle of Indifference. The wine/water paradox illustrates a flaw in that approach. The Two Envelopes problem illustrates another flaw. The "Grandfather" in the story argued that we can consider the problem specified if we assume that "the set of all distinct possible states" are uniformly distributed -- which is reasonable and correct when solving problems in quantum mechanics but unreasonable in ordinary everyday situations. Thinking along those lines, you can construct a set of all possible concentrations: (0 wine particles, N water particles), (1 wine particle, N-1 water), (2, N-2), ... (N, 0). That led to my x/(1-x) formulation, which has the advantage of avoiding the paradox as stated, but I would not argue it's a well-founded result.
  - sjg007 8y ago
    Why is it not well founded? Here an abstract amount (so a state) seems more realistic measure than the ratio which is a function of wine and water.
    - bhk 8y ago
      For one thing, there could be many distinct quantum states which correspond to the same water/wine ratio, so uniformly-distributed quantum states might not correspond to uniformly-distributed concentrations. But more fundamentally, it seems a stretch to apply these quantum mechanical assumptions. One could interpret the question as "in all possible real-world scenarios in which such a test might be performed, how often would the concentration be in this range?" ... in which case one could consider all realistic models for how the mixture might have been created, and the likelihood of each of those actually being the case, and consider more and more things coming up with a more complete answer ... but one would never have a correct answer without actually stating some additional assumptions.
omazurov 8y ago
> water/wine = x/(1-x) > wine/water = (1-x)/x It's symmetrical but not "naturally" uniform. If we consider both quantities as independent and uniformly distributed over [0..1[, then 1:1 ratio becomes much more likely to occur than 1:1000000. I find that approach "the most natural". The probability P(water/wine < 2) is 7/8 in that case.
- bhk 8y ago
  I agree. There are a number of ways to model the process of creating the mixture that will avoid the water vs. wine bias that is the essence of the "paradox" as stated. The main lesson is that the Principal of Indifference is "not even wrong". (It's nonsense.)

by bhk 8y ago

sjg007 8y ago

So this is defining the random variable not as the ratio but as the implicit function that computes the ratio? So we are really looking at the density function of the hidden variable X? For the original problem it feels like the interval constraints on the ratio are causing the issue since they will be inverse. But maybe I am not clear on that...

bhk 8y ago
The problem is not completely specified. However, in the study of probability and statistics there is a tendency to charge ahead and offer answers to incompletely specified problems, using e.g. the Principle of Indifference. The wine/water paradox illustrates a flaw in that approach. The Two Envelopes problem illustrates another flaw. The "Grandfather" in the story argued that we can consider the problem specified if we assume that "the set of all distinct possible states" are uniformly distributed -- which is reasonable and correct when solving problems in quantum mechanics but unreasonable in ordinary everyday situations. Thinking along those lines, you can construct a set of all possible concentrations: (0 wine particles, N water particles), (1 wine particle, N-1 water), (2, N-2), ... (N, 0). That led to my x/(1-x) formulation, which has the advantage of avoiding the paradox as stated, but I would not argue it's a well-founded result.
- sjg007 8y ago
  Why is it not well founded? Here an abstract amount (so a state) seems more realistic measure than the ratio which is a function of wine and water.
  - bhk 8y ago
    For one thing, there could be many distinct quantum states which correspond to the same water/wine ratio, so uniformly-distributed quantum states might not correspond to uniformly-distributed concentrations. But more fundamentally, it seems a stretch to apply these quantum mechanical assumptions. One could interpret the question as "in all possible real-world scenarios in which such a test might be performed, how often would the concentration be in this range?" ... in which case one could consider all realistic models for how the mixture might have been created, and the likelihood of each of those actually being the case, and consider more and more things coming up with a more complete answer ... but one would never have a correct answer without actually stating some additional assumptions.

omazurov 8y ago

> water/wine = x/(1-x) > wine/water = (1-x)/x It's symmetrical but not "naturally" uniform. If we consider both quantities as independent and uniformly distributed over [0..1[, then 1:1 ratio becomes much more likely to occur than 1:1000000. I find that approach "the most natural". The probability P(water/wine < 2) is 7/8 in that case.

bhk 8y ago
I agree. There are a number of ways to model the process of creating the mixture that will avoid the water vs. wine bias that is the essence of the "paradox" as stated. The main lesson is that the Principal of Indifference is "not even wrong". (It's nonsense.)