Bayesian recovery of sinusoids from noisy data with parallel tempering

Abstract

This study deals with parameter estimation of sinusoids within a Bayesian framework, where inferences about the parameters require an evaluation of complicated high-dimensional integrals and a solution of multi-dimensional optimisation of their posterior probability density function (PDF) under a combination of different prior PDFs of parameters. In this context, the authors make an attempt to improve an efficient stochastic procedure based on a parallel tempering Markov chain Monte Carlo sampler with a proposal distribution whose width varies with a Cramer-Rao lower bound (CRLB), known as a lower limit on variance of any unbiased estimator. Its algorithm is coded in 'Mathematica', which provides a much flexible and efficient computer programming environment. Computer simulations are included to corroborate theoretical developments and to compare the estimator performance with the CRLB for different length of data sampling and signal-to-noise ratio (SNR) conditions. Therefore all simulations support its effectiveness and demonstrate its performance in terms of CRLB for sufficiently high-SNR and short data lengths.