There has been much interest, recently, in the use of Bayesian formulations for solving image correspondence problems. For the two-view stereo matching problem, typical Bayesian formulations model the disparity prior as a pairwise Markov random field (MRF). Approximate inference algorithms for MRFs, such as graph cuts or belief propagation, treat the stereo matching problem as a labelling problem yielding discrete valued disparity estimates. In this paper, we propose a novel robust Bayesian formulation based on the recently proposed kernel maximum likelihood (KML) estimation framework. The proposed formulation uses probability density kernels to infer the posterior probability distribution of the disparity values. We present an efficient iterative algorithm, which uses a variational approach to form a KML estimate from the inferred distribution. The proposed algorithm yields continuous-valued disparity estimates, and is provably convergent. The proposed approach is validated on standard stereo pairs, with known sub-pixel disparity ground-truth data.