On the generic nonconvergence of Bayesian actions and beliefs

Abstract

SupposeYn is a sequence of i.i.d. random variables taking values in Y, a complete, separable, non-finite metric space. The probability law indexed byθeΘ, is unknown to a Bayesian statistician with priorμ, observing this process. Generalizing Freedman [8], we show that “generically” (i.e., for a residual family of (θ,μ) pairs) the posterior beliefs do not weakly converge to a point-mass at the “true”θ. Furthermore, for every open setG ⊂Θ, generically, the Bayesian will attach probability arbitrarily close to one toG infinitely often. The above result is applied to a two-armed bandit problem with geometric discounting where armk yields an outcome in a complete, separable metric spaceYk. If the infimum of the possible rewards from playing armk is less than the infimum from playing armk', then armk is (generically) chosen only finitely often. If the infimum of the rewards are equal, then both arms are played infinitely often.