Bayesian revolution in spectral analysis

Abstract

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. IntroductionScience is all about identifying and understanding organized structures or patterns in nature. In this regard periodic patterns have proven especially important. Nowhere is this more evident than in the field of astronomy. Periodic phenomena allow us to determine fundamental properties like mass and distance, enable us to probe the interior of star through the new techniques of stellar seismology, detect new planets and discover exotic states of matter like neutron stars and black holes. Clearly any fundamental advance in our ability to detect periodic phenomena will have profound consequences in our ability to unlock nature's secrets. The purpose of this article is to describe such an advance and provide illustrations of its power through several examples in physics and astronomy.This advance is based on a theory of extended logic called Bayesian probability theory (BPT) or Bayesian Inference. It is not the purpose of this paper to present the foundations of Bayesian inference. The reader is referred to the work of E. T. Jaynes 1 . For a brief introduction to the subject see review articles by Loredo [19] †