Scale Mixtures of Complex Gaussians and Bayesian Shrinkage

Complex-valued distributions are widely used in fields such as signal processing and neuroimaging, where magnetic resonance imaging (MRI) and functional MRI (fMRI) data are inherently complex-valued due to phase imperfections. Leveraging full complex-valued data improves statistical power, inference, and prediction compared to using only magnitude or real-valued subsets. This paper extends scale mixtures of Gaussians to the complex domain, deriving the most general complex-valued versions of Student-t, Laplace, and GDP distributions and their real-valued equivalents as special cases. We apply these distributions as shrinkage priors in complex Bayesian regression, developing novel MCMC algorithms that estimate correlations between real and imaginary components. Simulations and fMRI data demonstrate that complex-valued shrinkage priors enhance variable selection, coefficient estimation, and predictive accuracy, particularly when real and imaginary parts are highly correlated. The R package cplxrv provides tools for simulating complex variables and implementing the proposed MCMC methods.