Analytical and Numerical Investigations of Probabilistic Monochromatic Problem

Abstract

A probabilistic formalism, relying on Bayes’ theorem and linear Gaussian inversion, is adapted, so that a monochromatic problem can be investigated. The formalism enables an objective test in probabilistic terms of the quantities and model concepts involved in the problem at hand. With this formalism, an amplitude (linear parameter), a frequency (non-linear parameter) and a hyperparameter of the Gaussian amplitude prior are inferred jointly given simulated data sets with Gaussian noise contributions. For the amplitude, an analytical normal posterior follows which is conditional on the frequency and the hyperparameter. The remaining posterior estimates the frequency with an uncertainty of MHz, while the convolution of a standard approach would achieve an uncertainty of some GHz. This improvement in the estimation is investigated analytically and numerically, revealing for instance the positive effect of a high signal-to-noise ratio and/or a large number of data points. As a fixed choice of the hyperparameter imposes certain results on the amplitude and frequency, this parameter is estimated and, thus, tested for plausibility as well. From abstract point of view, the model posterior is investigated as well.