Indifference, neutrality and informativeness: generalizing the three prisoners paradox

Abstract.

The uniform prior distribution is often seen as a mathematical description of noninformativeness. This paper uses the well-known Three Prisoners Paradox to examine the impossibility of maintaining noninformativeness throughout hierarchization. The Paradox has been solved by Bayesian conditioning over the choice made by the Warder when asked to name a(nother) prisoner who will be shot. We generalize the paradox to situations of N prisoners, k executions and m announcements made by the Warder. We then extend the consequences of hierarchically placing uniform and symmetrical priors (for example in the classical N = 3, k = 2, m = 1 scenario) for the probability p of the Warder naming Prisoner B, say. We prove that breaks of indifference and neutrality caused by assignment of uniform and symmetrical priors in lieu of degenerate indifference probabilities hold in general. Speaking of unknown probabilities or of probability of a probability must be forbidden as meaningless. Bruno de Finetti, 1977 I regard the use of hierarchical chains as a technique helping you to sharpen your subjective probabilities. I. J. Good, 1981