Asymptotics of Bayesian Inference for a Class of Probabilistic Models under Misspecification
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Abstract
In this paper, sequential prediction is studied. The typical assumptions about the probabilistic model in sequential prediction are following two cases. One is the case that a certain probabilistic model is given and the parameters are unknown. The other is the case that not a certain probabilistic model but a class of probabilistic models is given and the parameters are unknown. If there exist some parameters and some models such that the distributions that are identified by them equal the source distribution, an assumed model or a class of models can <i>represent</i> the source distribution. This case is called that <i>specifiable condition</i> is satisfied. In this study, the decision based on the Bayesian principle is made for a class of probabilistic models (not for a certain probabilistic model). The case that <i>specifiable condition</i> is not satisfied is studied. Then, the asymptotic behaviors of the cumulative logarithmic loss for individual sequence in the sense of almost sure convergence and the expected loss, i.e. redundancy are analyzed and the constant terms of the asymptotic equations are identified.
Journal

 IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E97.A(12), 23522360, 2014
The Institute of Electronics, Information and Communication Engineers