Bayesian Analysis V : Amplitude Estimation , Multiple Well Separated

Bayesian probability theory is used to estimate the amplitude of a single exponentially decaying sinusoid in NMR free induction decay (FID) data. The posterior probability for the amplitude is derived independent of the phase, frequency, decay rate constant, and variance of the noise. The estimate is shown to be accurate and precise in the sense that as the noise approaches zero, the estimate approaches the true value of the amplitude and the uncertainty in the estimate approaches zero. The uncertainty in the estimate is shown to varying inversely with the square root of the sampling rate. Finally, the calculation is applied in two examples. The rst example demonstrates probability theory's ability to estimate frequencies and amplitudes in very low signal-to-noise. The second example illustrates the use of this calculation when the data contain multiple, well-separated sinusoids.