Principle of transformation groups
The principle of transformation groups is a rule for assigning epistemic probabilities in a statistical inference problem. It was first suggested by Edwin T. Jaynes and can be seen as a generalisation of the principle of indifference.
This can be seen as a method to create objective ignorance probabilities in the sense that two people who apply the principle and are confronted with the same information will assign the same probabilities.
The method is motivated by the following normative principle, or desideratum:
In two problems where we have the same prior information we should assign the same prior probabilities
The method then comes about from "transforming" a given problem into an equivalent one. This method has close connections with group theory, and to a large extent is about finding symmetry in a given problem, and then exploiting this symmetry to assign prior probabilities.
In problems with discrete variables (e.g. dice, cards, categorical data) the principle reduces to the principle of indifference, as the "symmetry" in the discrete case is a permutation of the labels, that is the permutation group is the relevant transformation group for this problem.
In problems with continuous variables, this method generally reduces to solving a differential equation. Given that differential equations do not always lead to unique solutions, this method cannot be guaranteed to produce a unique solution. However, in a large class of the most common types of parameters it does lead to unique solutions (see the examples below)
Consider a problem where all you are told is that there is a coin, and it has a head (H) and a tail (T). Denote this information by I. You are then asked "what is the probability of Heads?". Call this problem 1 and denote the probability P(H|I). Consider another question "what is the probability of Tails?". Call this problem 2 and denote this probability by P(T|I).
Now from the information which was actually in the question, there is no distinction between heads and tails. The whole paragraph above could be re-written with "Heads" and "Tails" interchanged, and "H" and "T" interchanged, and the problem statement would not be any different. Using the desideratum then demands that
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