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Large deviations and the Bayesian estimation of higher-order Markov transition functions

Part of: Limit theorems Inference from stochastic processes Markov processes Parametric inference

Published online by Cambridge University Press:  14 July 2016

F. Papangelou
F. Papangelou*
Affiliation:
University of Manchester
*
∗Postal address: Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK.
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Abstract

In the Bayesian estimation of higher-order Markov transition functions on finite state spaces, a prior distribution may assign positive probability to arbitrarily high orders. If there are n observations available, we show (for natural priors) that, with probability one, as n → ∞ the Bayesian posterior distribution ‘discriminates accurately' for orders up to β log n, if β is smaller than an explicitly determined β0. This means that the ‘large deviations' of the posterior are controlled by the relative entropies of the true transition function with respect to all others, much as the large deviations of the empirical distributions are governed by their relative entropies with respect to the true transition function. An example shows that the result can fail even for orders β log n if β is large.

Keywords

MARKOV CHAINSHIGHER-ORDER TRANSITION FUNCTIONSRELATIVE ENTROPYLARGE DEVIATIONSPOSTERIOR DISTRIBUTIONSEXPONENTIAL DECAY

MSC classification

Primary: 60F10: Large deviations 60F15: Strong theorems
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 62F15: Bayesian inference 62M05: Markov processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 18 - 27
Copyright
Copyright © Applied Probability Trust 1996 

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