Bayesian analysis. III. Applications to NMR signal detection, model selection, and parameter estimation

The two preceding articles developed the application of Bayesian probability theory to the problems of parameter estimation, signal detection, and model selection on quadrature NMR data in some generality. Here those procedures are used to analyze free induction decay data, when the models are sinusoidal. The exact relationship between Bayesian probability theory and the discrete Fourier-transform power spectrum is derived, and it is shown that the discrete Fourier-transform power spectrum is an optimal frequency estimator for a wide class of problems. Signal detection and model selection problems are then examined, and examples are given that demonstrate the ability of Bayesian probability theory to determine the best model of a process even when more complex models fit the data better.