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Transformation Groups

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Transformation Groups

H-index: 89 CiteScore: 3. Journal of Numerical Mathematics. H-index: 26 CiteScore: 2. H-index: 59 CiteScore: 3. H-index: 91 CiteScore: 3.

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H-index: 56 CiteScore: 2. H-index: 31 CiteScore: 2. H-index: 71 CiteScore: 2. H-index: 34 CiteScore: 1. Abstract and Applied Analysis. Boundary Value Problems. Write a review. Thus we have a unique solution. This can be alternatively phrased as: a person who does not use the principle of indifference to assign prior probabilities to discrete variables, is either not ignorant about them, or reasoning inconsistently. This is the easiest example for continuous variables.

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It is given by stating one is "ignorant" of the location parameter in a given problem. Examples of location parameters include mean parameter of normal distribution with known variance and median parameter of Cauchy distribution with known inter-quartile range. This is because of the relation:. So simply "shifting" all quantities up by some number b and solving in the "shifted space" and then "shifting" back to the original one should give exactly the same answer as if we just worked on the original space.

Where, as before f. Examples include the standard deviation of a normal distribution with known mean, the gamma distribution. The "symmetry" in this problem is found by noting that. But, unlike in the location parameter case, the Jacobian of this transformation in the sample space and the parameter space is a , not 1. Which is invariant i.

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Which is the well-known Jeffreys prior for scale parameters, which is "flat" on the log scale, although it is derived using a different argument to that here, based on the Fisher information function. The fact that these two methods give the same results in this case does not imply it in general. Edwin Jaynes used this principle to provide a resolution to Bertrand's Paradox [2] by stating his ignorance about the exact position of the circle. The details are available in the reference or in the link.

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This argument depends crucially on I ; changing the information may result in a different probability assignment. It is just as crucial as changing axioms in deductive logic - small changes in the information can lead to large changes in the probability assignments allowed by "consistent reasoning". To illustrate suppose that the coin flipping example also states as part of the information that the coin has a side S i.

Denote this new information by N. The same argument using "complete ignorance", or more precisely, the information actually described, gives:. But this seems absurd to most people - intuition tells us that we should have P S very close to zero. This is because most people's intuition do not see "symmetry" between a coin landing on its side compared to landing on heads. Our intuition says that the particular "labels" actually carry some information about the problem.

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A simple argument could be used to make this more formal mathematically e. It could reasonably be assumed that:. Note that this new information probably wouldn't break the symmetry between "heads" and "tails", so that permutation would still apply in describing "equivalent problems", and we would require:. This is a good example of how the principle of transformation groups can be used to "flesh out" personal opinions.

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  • All of the information used in the derivation is explicitly stated. If a prior probability assignment doesn't "seem right" according to what your intuition tells you, then there must be some "background information" which has not been put into the problem. In some sense, by combining the method of transformation groups with one's intuition can be used to "weed out" the actual assumptions one has. This makes it a very powerful tool for prior elicitation.

    Introducing the thickness of the coin as a variable is permissible because its existence was implied by being a real coin but its value was not specified in the problem. Introducing a "nuisance parameter" and then making the answer invariant to this parameter is a very useful technique for solving supposedly "ill-posed" problems like Bertrand's Paradox.

    Introduction to Compact Transformation Groups, Volume 46 - 1st Edition

    This has been called "the well-posing strategy" by some. The real power of this principle lies in its application to continuous parameters, where the notion of "complete ignorance" is not so well defined as in the discrete case. However, if applied with infinite limits, it often gives improper prior distributions.