３囚人問題等に関するベイズの解

抄録

 More than two decades ago, vos Savant lighted a fuse to fight against those, who believe an event of winning a prize on the Monty Hall’s host-guest TV show takes place equally likely, and so that a guest cannot help falling into a dilemma whether to stay with his or her first choice or to switch it to the other left. At that time, she claimed to them that the guest had better switch it because a subjective probability of winning the prize became from one third to two thirds. Nowadays, much of the literature is apt to accept her claim due to a seemingly Bayesian-like decision tree with second branches for arbitrary conditional probabilities. However, it is notorious that her solution come from the tree is too counterintuitive to coherently revise subjective probabilities of the prize whereabouts.

  Instead of making such an arbitrary decision, therefore, we would like to show in this paper that a problem like the Monty Hall, three prisoners, and so on has a trinomial distribution, whose parameters depend upon the subjective probabilities of the whereabouts. A conjugate Dilicilet distribution is then used for not only prior but also posterior information of the probabilities. So, we study here how a Bayes’ solution, derived from a minimization problem of a posterior loss function, is able to rationally and coherently revise subjective probabilities as well as conditional ones.