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Analytical and Numerical Investigations of Probabilistic Monochromatic Problem

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Document pages: 16 pages

Abstract: A probabilistic formalism, relyingon Bayes’ theorem and linear Gaussian inversion, is adapted, so that amonochromatic problem can be investigated. The formalism enables an objectivetest in probabilistic terms of the quantities and model concepts involved inthe problem at hand. With this formalism, an amplitude (linear parameter), afrequency (non-linear parameter) and a hyperparameter of the Gaussian amplitudeprior are inferred jointly given simulated data sets with Gaussian noisecontributions. For the amplitude, an analytical normal posterior follows whichis conditional on the frequency and the hyperparameter. The remaining posteriorestimates the frequency with an uncertainty of MHz, while the convolution of astandard approach would achieve an uncertainty of some GHz. This improvement inthe estimation is investigated analytically and numerically, revealing forinstance the positive effect of a high signal-to-noise ratio and or a largenumber of data points. As a fixed choice of the hyperparameter imposes certainresults on the amplitude and frequency, this parameter is estimated and, thus,tested for plausibility as well. From abstract point of view, the model posterior isinvestigated as well.

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