Asymptotics of Bayesian Inference for a Class of Probabilistic Models under Misspecification

Nozomi MIYA  Tota SUKO  Goki YASUDA  Toshiyasu MATSUSHIMA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E97-A   No.12   pp.2352-2360
Publication Date: 2014/12/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E97.A.2352
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Prediction
a class of probabilistic models,  Bayesian inference,  cumulative logarithmic loss,  misspecification,  sequential prediction,  

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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 represent the source distribution. This case is called that specifiable condition 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 specifiable condition 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.