  We propose a regression model for non-central $\chi$ (NC-$\chi$) distributed
functional magnetic resonance imaging (fMRI) and diffusion weighted imaging
(DWI) data, with the heteroscedastic Rician regression model as a prominent
special case. The model allows both parameters in the NC-$\chi$ distribution to
be linked to explanatory variables, with the relevant covariates automatically
chosen by Bayesian variable selection. A highly efficient Markov chain Monte
Carlo (MCMC) algorithm is proposed for simulating from the joint Bayesian
posterior distribution of all model parameters and the binary covariate
selection indicators. Simulated fMRI data is used to demonstrate that the
Rician model is able to localize brain activity much more accurately than the
traditionally used Gaussian model at low signal-to-noise ratios. Using a
diffusion dataset from the Human Connectome Project, it is also shown that the
commonly used approximate Gaussian noise model underestimates the mean
diffusivity (MD) and the fractional anisotropy (FA) in the single-diffusion
tensor model compared to the theoretically correct Rician model.
