We present a learning-based selection argument for Linear Bayesian Nash equilibrium in a Walrasian auction. Endowments vary stochastically; traders model residual supply as linear, estimate its slope from past trade data, and periodically update these estimates. With quadratic preferences, this learning process converges to the unique LBN. In an example with non-quadratic preferences, it converges to a steady state close to a particular equilibrium of the corresponding deterministic setting; strategies played are not an equilibrium, but utility sacrificed is negligible. Anonymity and statistical learning therefore support use of LBN under quadratic utility, and motivate a related concept under non-quadratic utility.