The idea that equivalent problems must yield the same result is what Jaynes calls the principle of maximum ignorance. To Jaynes, problems are equivalent if they only differ in respect to factors not mentioned in the statement of the problem, and we cannot allow the variation of factors not specified in the problem to alter our result - our result needs to be general enough to cover all the different situations the problem could be describing. Jaynes' method for ensuring this via invariance is called the principle of transformation groups (which could be a good topic for a future article). However, his critics, namely Darrell Rowbottom, Nicholas Shackel and Diederik Aerts, note that whilst this method tries to ensure that the principle of indifference i.e. the uniform distribution is used correctly and applied to the right random variable, the principle of maximum ignorance is itself using the principle of indifference – not on the level of random variables, but on the level of equivalent problems. He is effectively using a principle to help ensure the correct use of that same principle. The worry is that any problems associated with applying the principle of indifference on the level of random variables might still persist when used on the higher level of equivalent problems. ‘with most other writers on probability theory that it is dangerous to apply this principle at the level of indifference between events, because our intuition is a very unreliable guide in such matters, as Bertrand’s paradox illustrates. However, the principle of indifference may, in [his] view, be applied legitimately at the more abstract level of indifference between problems; because that is a matter that is definitely determined by the statement of a problem, independently of our intuition.’ It does therefore seem like a less risky application of the principle of indifference, but by no means fool proof – how can you be sure that you have included all the relevant invariances? The statement of a problem will always leave a great many things unstated, and whilst some of them have no bearing on the solution, for example the time of day or the strength of the dollar, there is always a risk that we have not accounted for all of the relevant factors. This bears similarity to the problems faced by scientists in trying to perform a “fair test” when it is hard to guarantee that they are controlling (or keeping constant) all but the independent and dependent variables. ‘on the one hand, one cannot deny the force of arguments which, by pointing to such things as Bertrand’s paradox, demonstrate the ambiguities and dangers in the principle of indifference. But on the other hand, it is equally undeniable that use of this principle has, over and over again, led to correct, nontrivial, and useful predictions. Thus it appears that while we cannot wholly accept the principle of indifference, we cannot wholly reject it either; to do so would be to cast out some of the most important and successful applications of probability theory.’ As previously mentioned, Maxwell was responsible for one of the first great triumphs of kinetic theory in which he was able to predict various macroscopic properties of gases such as viscosity, thermal conductivity, diffusion rates etc. from information that seemed inadequate to determine these states uniquely. He ‘was able to predict all these quantities correctly by a ‘pure thought’ probability analysis which amounted to recognizing the ‘equally possible’ cases.’ Because Maxwell’s theory leads to testable predictions, the question of him applying the uniform distribution correctly does not belong to the realm of philosophy but to the realm of verifiable fact. Whilst this is offered as a defense by Jaynes, it also demands that these successes be explained by Jaynes’ new principles. New theories must to be able to explain the past successes of old theories. Jaynes acknowledges this, saying that he agrees A second reassurance offered by Jaynes is that
To this end Jaynes suggests
“that the cases in which the principle of indifference has been applied successfully in the past are just the ones in which the solution can be “reverbalized” so that the actual calculations used are seen as an application of indifference between problems, rather than events.”
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