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.
Highlights
Fourier transformation tools are used to obtain information about spectra for a given data set
As any data has an uncertainty, Fourier transformation techniques can be supported by probabilistic theory captured by Bayes’ theorem [1] to improve scientific results and conclusions
After a basic formalism has been derived, it could be shown in Ref. [4] that the band limitation can be well inferred from experimental data
Summary
Fourier transformation tools are used to obtain information about spectra for a given data set. As any data has an uncertainty, Fourier transformation techniques can be supported by probabilistic theory captured by Bayes’ theorem [1] to improve scientific results and conclusions. The spectral band limits and the uncertainty on the derived quasi-continuous spectrum, originating in non-probed Fourier coefficients, have been inferred jointly. From scientific point of view, any analysis scheme should be tested given simulated noisy data for which all model assumptions are clear. Analysis results and model assumptions can be investigated which is achieved objectively by a probabilistic ansatz. If this can be carried out analytically, valuable information is available when only actual measured data are given for a scientific problem.
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