Matched field processing (MFP) compares the measures to the modeled pressure fields received at an array of sensors to localize a source in an ocean waveguide. Typically, there are only a few sources when compared to the number of candidate source locations or range-depth cells. We use sparse Bayesian learning (SBL) to learn a common sparsity profile corresponding to the location of present sources. SBL performance is compared to traditional processing in simulations and using experimental ocean acoustic data. Specifically, we localize a quiet source in the presence of a surface interferer in a shallow water environment. This multi-frequency scenario requires adaptive processing and includes modest environmental and sensor position mismatch in the MFP model. The noise process changes likely with time and is modeled as a non-stationary Gaussian process, meaning that the noise variance changes across snapshots. The adaptive SBL algorithm models the complex source amplitudes as random quantities, providing robustness to amplitude and phase errors in the model. This is demonstrated with experimental data, where SBL exhibits improved source localization performance when compared to the white noise gain constraint (-3 dB) and Bartlett processors.