Bayesian inference and Gibbs sampling in spectral analysis and parameter estimation. I

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Abstract
Bayesian inference theory and Gibbs sampling techniques are introduced and applied to spectral analysis and parameter estimation for both single- and multiple-frequency signals. Specifically, the marginal posterior probabilities for amplitudes and frequencies are obtained by using Gibbs sampling without performing the integrations, no matter whether the variance of the noise is known or unknown. The best estimates of the parameters can be inferred from these probabilities together with the corresponding variances. In addition, when the variance of the noise is unknown, an estimate about the variance of the noise can also be made. Comparisons of our results have been made with results using the FFT method as well as with Bretthorst's (1990) method. The approach outlined shows several advantages.