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The boundary mode Laplace approximation to the Bayes factor was utilized for testing extrabinomial variation. The chance of forming a boundary mode was high when the boundary mode Laplace approximation to the Bayes factor occurred at the boundary of the parameter space. The half-normal density function and the exponential density function can also be utilized for boundary mode Laplace approximations depending on the difference of the boundary mode from the true value., 1. INTRODUCTION For testing or choosing between two competing hypotheses [H.sub.0] and [H.sub.1], the Bayes factor measures the strength of evidence provided by the data y. The Bayes factors (B) [...]

Prior assumptions under the null and alternative hypotheses are required to compute a Bayes factor for testing a nested hypotheses in the presence of a nuisance parameter. It is shown that the Schwartz criterion can give a reasonably approximate solution to a Bayesian testing problem. In certain cases, the log of the Bayes factor can be approximated by the Schwartz criterion with an acceptable degree of error.

The Laplace method as an analytical tool for estimating expected values of functions of parameters and marginal densities in the Bayesian context is presented. Specifically, asymptotic expansions for posterior expectations of functions of parameters have been derived when the mode is on the edge of the parametric space. Second-order accuracy is obtained even if the mode is situated on the boundary.

The problem of extracting sinusoid signals from noisy observations made at equally spaced times is considered. Eigenanalysis methods, such as Pisarenko's method and the extended Prony method, find the eigenvector [...]