Upper Bounds for Symmetric Markov Transition Functions.

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.

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

Journal of Applied Probability ◽

10.1017/s0021900200103687 ◽

1996 ◽

Vol 33 (01) ◽

pp. 18-27

Author(s):

F. Papangelou

Keyword(s):

Large Deviations ◽

Bayesian Estimation ◽

Transition Function ◽

Higher Order ◽

Positive Probability ◽

State Spaces ◽

Empirical Distributions ◽

Transition Functions ◽

Markov Transition ◽

Finite State

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 β 0. 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.

Download Full-text