AN ASYMPTOTIC THEORY OF BAYESIAN-INFERENCE FOR TIME-SERIES

This paper develops an asymptotic theory of Bayesian inference for tim e series. A limiting representation of the Bayesian data density is ob tained and shown to be of the same general exponential form for a wide class of likelihoods and prior distributions. Continuous time and dis crete time cases are studied. In discrete time, an embedding theorem i s given which shows how to embed the exponential density in a continuo us time process. From the embedding we obtain a large sample approxima tion to the model of the data that corresponds to the exponential dens ity. This has the form of discrete observations drawn from a nonlinear stochastic differential equation driven by Brownian motion. No assump tions concerning stationarity or rates of convergence are required in the asymptotics. Some implications for statistical testing are explore d and we suggest tests that are based on likelihood ratios (or Bayes f actors) of the exponential densities for discriminating between models .