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Wine-water paradox leaks


The wine-water paradox is a hoax. Note that the wine-water paradox is supposedly a way of demonstrating that the principle of insufficient reason does not contain sufficient information to choose a prior decision in Bayesian hypothesis testing. The canonical presentation relies on the ratio of wine to water in a mixed solution, but it makes a fatal error in choosing a uniform probability distribution over the interval of the quotient, rather than the logarithm of the quotient. Taking the logarithms of the ratio resolves the paradox.

eqn3910

eqn3911

Givens:

  1. r = wine / water
  2. s = water / wine = 1 / r
  3. 1/3 <= r, s <= 3

Paradox (false):

  • P(r <=2) = (2-1/3) / (3-1/3) = 5/8
  • P(s >=1/2) = (3-1/2) / (3-1/3) = 15/16

Resolution using logarithms:

  • P(r <= 2) =  (log 2 – log 1/3) / (log 3 – log 1/3)
  • P(s >= 1/2) = (log 3 – log 1/2) / (log 3 – log 1/3)
  • These are equal. Paradox resolved.

This apparent paradox is just a numerical artifact of an erroneous assumption about the uniform probability distribution of a quotient.

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About Ben Mazzotta

Ben Mazzotta is a postdoc at the Center for Emerging Market Enterprises (CEME). His study of the Cost of Cash is part of CEME's research into inclusive growth.

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