Data from magnetic resonance imaging (MRI) modalities are acquired in terms of spatial frequencies, and are therefore representable in Fourier or k-space. The data are then typically inverse-Fourier-transformed, and the magnitude of the complex signal at each location is extracted to display an image. Bayesian image analysis can improve image quality, by balancing a priori expectations of image characteristics with a model for the noise process via Bayes Theorem. In contrast to conventional Bayesian image analysis, we here preform Bayesian image analysis in Fourier space (BIFS), i.e., directly applied to the raw MRI data in Fourier space. By specifying the Bayesian model in Fourier space, spatially correlated priors, that are relatively difficult to model and compute in conventional image space, can be efficiently modeled as a set of independent processes across Fourier space; the priors in Fourier space are modeled as independent, but tied together by defining a function over Fourier space for the parameters. We will describe the BIFS modeling approach specifically for MRI, and demonstrate benefits to model specification, posterior image properties, and computational efficiency.