Complex-valued Bayesian parameter estimation via Markov chain Monte Carlo

The study of parameter estimation of a specified model has a long history. In statistics, Bayesian analysis via Markov chain Monte Carlo (MCMC) sampling is an efficient way for parameter estimation. However, the existing MCMC sampling is only performed in the real parameter space. In some situation, complex-valued parametric modeling is more preferable as complex representation brings economies and insights that would not be achieved by real-valued representation. Therefore, to estimate complex-valued parameters, it is more convenient and elegant to perform the MCMC sampling in the complex parameter space. In this paper, firstly, based on the assumption that the observation signal is proper, two complex MCMC algorithms using the Metropolis–Hastings sampling and the differential evolution are proposed, in which the probability density functions (pdfs) in Bayesian estimation are characterized by the usual Hermitian covariance matrices. Secondly, to improve the performance for the case that the observation signal is improper, two augmented complex MCMC algorithms are developed, where the pdfs are computed by the augmented complex statistics. Both theoretical studies and numerical simulations are presented to show the effectiveness of the proposed algorithms in complex-valued parameter estimation.