But is it really an inconsistency?
An inconsistency?
Keynes gives a striking example of what he regards as a fatal flaw in the Principle of Indifference.
Consider the definition of specific volume: v = 1/ρ = V/m. That is, the density ρ is the inverse of the specific volume v.
Suppose all we can say about a container is that [Case A] the specific volume lies between 1 and 3 ([cubic somethings]/mass). Absent other information the Laplacian principle would say to us that it is equally probable for the exact specific volume to lie in the interval [1,2] as for it to lie in [2,3].
Now if we take the inverses [Case B] of the interval limits, we obtain [1, 1/2] and [1/2, 1/3]. As we have already distinguished the probabilities for the specific volume, it seems that the density probabilities are forced at 0.5 chance of finding the density in [1, 1/2] and 0.5 chance of finding it in [1/3, 1/2].
But had we begun with the interval of [1, 1/2] for the density, we would say the sub-intervals corresponding to equiprobability are [1, 1/4] and [1/4, 1/2], as these distances are equal.
We know the inverse function of every element of the domain yields an element of the range. So for [1,2], the range is [1,1/2] and for [2,3] the range is [1/2, 1/3]. Unlike Case A, these Case B sets are of unequal measure.
Keynes arrives at this incongruity by a slightly different route than the one I use. He thinks the principle implies that the "specific density" (sic) is as likely to be found "between 1 and 2/3 as between 2/3 and 1/3." Since the point set [1/3,2/3] is a proper subset of [1,2/3] the claim is clearly false, which is his point. In other words, he means 1/3[1,3] --> [1,2/3],[2/3,1/3].
Anyway, "Specific volume and specific density [sic] are simply alternative methods of measuring the same objective quantity; and there are many methods which might be adopted, each yielding on application of the Principle of Indifference a different probability for a given objective variation in the quantity."
Keynes here makes a strong argument against the Laplacian/Bayesian outlook.
That is,
Case A ¬≡ Case B
A response might be that the notion that Case A's intervals directly correspond with Case B's is proved to be incorrect, but that means that the intervals for Case A actually correspond with the second set of intervals for Case B, which have the same measure as those of Case A.
Perhaps the issue is that such a rule should have been made explicit, which evidently it had not yet been. That is, the principle says that [1,2] and [2,3] present equiprobabilities because the distances are equal. Hence the reciprocal interval's distance must be the largest possible, given the data, which in this case is [1,2/3] and the reciprocal interval must be divided exactly in half in order for the principle's equiprobability assumption to apply.
Keynes however does give potent examples of "geometrical probability" whereby the principle can seemingly be applied equally well to what seem to be equivalent scenarios, with the results not being the same.
(See pages 48 through 56 of A Treatise on Probability.)
Keynes gives a striking example of what he regards as a fatal flaw in the Principle of Indifference.
Consider the definition of specific volume: v = 1/ρ = V/m. That is, the density ρ is the inverse of the specific volume v.
Suppose all we can say about a container is that [Case A] the specific volume lies between 1 and 3 ([cubic somethings]/mass). Absent other information the Laplacian principle would say to us that it is equally probable for the exact specific volume to lie in the interval [1,2] as for it to lie in [2,3].
Now if we take the inverses [Case B] of the interval limits, we obtain [1, 1/2] and [1/2, 1/3]. As we have already distinguished the probabilities for the specific volume, it seems that the density probabilities are forced at 0.5 chance of finding the density in [1, 1/2] and 0.5 chance of finding it in [1/3, 1/2].
But had we begun with the interval of [1, 1/2] for the density, we would say the sub-intervals corresponding to equiprobability are [1, 1/4] and [1/4, 1/2], as these distances are equal.
We know the inverse function of every element of the domain yields an element of the range. So for [1,2], the range is [1,1/2] and for [2,3] the range is [1/2, 1/3]. Unlike Case A, these Case B sets are of unequal measure.
Keynes arrives at this incongruity by a slightly different route than the one I use. He thinks the principle implies that the "specific density" (sic) is as likely to be found "between 1 and 2/3 as between 2/3 and 1/3." Since the point set [1/3,2/3] is a proper subset of [1,2/3] the claim is clearly false, which is his point. In other words, he means 1/3[1,3] --> [1,2/3],[2/3,1/3].
Anyway, "Specific volume and specific density [sic] are simply alternative methods of measuring the same objective quantity; and there are many methods which might be adopted, each yielding on application of the Principle of Indifference a different probability for a given objective variation in the quantity."
Keynes here makes a strong argument against the Laplacian/Bayesian outlook.
That is,
Case A ¬≡ Case B
A response might be that the notion that Case A's intervals directly correspond with Case B's is proved to be incorrect, but that means that the intervals for Case A actually correspond with the second set of intervals for Case B, which have the same measure as those of Case A.
Perhaps the issue is that such a rule should have been made explicit, which evidently it had not yet been. That is, the principle says that [1,2] and [2,3] present equiprobabilities because the distances are equal. Hence the reciprocal interval's distance must be the largest possible, given the data, which in this case is [1,2/3] and the reciprocal interval must be divided exactly in half in order for the principle's equiprobability assumption to apply.
Keynes however does give potent examples of "geometrical probability" whereby the principle can seemingly be applied equally well to what seem to be equivalent scenarios, with the results not being the same.
(See pages 48 through 56 of A Treatise on Probability.)
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