Separable nonlinear inverse problems arise in many applications where a forward model depends linearly on some unknowns and nonlinearly on others, including semi-blind deconvolution and kernel learning. In this talk, we adopt a Bayesian framework and assume Gaussian noise, along with Gaussian or other structured priors on the linear variables, as appropriate. We examine several prior models for the nonlinear parameters and show that these choices yield maximum a posteriori estimates formulated as solutions to regularized separable nonlinear least-squares problems that can be efficiently solved using the variable projection method. In addition, by marginalizing the linear variables, we obtain a reduced posterior over the nonlinear parameters that provides a Bayesian interpretation of variable projection, improves computational efficiency, and enables systematic uncertainty quantification in kernel learning. This is Joint work with Jordan Dworaczyk