Bayesian revolution in spectral analysis

Abstract

The discrete Fourier transforms (DFT) is ubiquitous in spectral analysis as a result of the introduction of the Fast Fourier transform by Cooley and Tukey in 1965. In 1987, E. T. Jaynes derived the DFT using Bayesian Probability Theory and provided surprising new insights into its role in spectral analysis. From this new perspective the spectral resolution achievable is directly dependent on the signal to noise ratio and can be orders of magnitude better than that of a conventional Fourier power spectrum or periodogram. This was the starting point for an ongoing Bayesian revolution in spectral analysis which is reviewed in this paper, with examples taken from physics and astronomy. The revolution is based on a viewpoint of Bayesian Inference as extended logic.