TITLE:
Analytical and Numerical Investigations of Probabilistic Monochromatic Problem
AUTHORS:
Stefan Schmuck, Jakob Svensson
KEYWORDS:
Bayes’ Theorem, Linear Inversion, Uncertainty, Monochromatic, Spectral Limitation
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.7 No.4,
April
16,
2019
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.