The Art of Probability Assignment

Abstract: The problem of assigning probabilities when little is known is analized in the case where the quanities of interest are physical observables, i.e. can be measured and their values expressed by numbers. It is pointed out that the assignment of probabilities based on observation is a process of inference, involving the use of Bayes' theorem and the choice of a probability prior. When a lot of data is available, the resulting probability are remarkably insensitive to the form of the prior. In the oposite case of scarse data, it is suggested that the probabilities are assigned such that they are the least sensitive to specific variations of the probability prior. In the continuous case this results in a probability assignment rule wich calls for minimizing the Fisher information subject to constraints reflecting all available information. In the discrete case, the corresponding quantity to be minimized turns out to be a Renyi distance between the original and the shifted distribution.