Dice have no memories, but I do: a defence of the reverse gambler's belief

Abstract

We investigate the problem of predicting the outcomes of a sequence of discrete random variables that are almost uniform, in the sense that they are generated from a random process that is designed to produce independent uniform outcomes but may not do so exactly. Using assumptions based around this notion we derive a useful stochastic ordering. We reject the gambler’s belief as unsound and find that the reverse gambler’s belief is the optimal prediction method. This method arises under a wide class of Bayesian models. One of the main contributions of this paper is that it uses only weak and intuitive prior assumptions and should therefore be more palatable to sceptics than existing Bayesian models. Show more