Abstract: A seminal work by Moulin (1980) shows that the median voting scheme fully characterizes (deterministic) strategy-proof facility location mechanism for single-peaked preferences. In this simple setting, median also achieves the optimal social cost. In $d$ dimensions, strategy-proof mechanism is characterized by coordinate-wise median, which is known to have a large $\sqrt{d}$ approximation ratio of the social cost in the Euclidean space, whereas the socially optimal mechanism fails at being strategy-proof. In light of the negative results in the classic, worst-case setting, we initiate the study of Bayesian mechanism design for strategy-proof facility location for multi-dimensional Euclidean preferences, where the agents' preferences are drawn from a distribution. We approach the problem via connections to algorithmic high-dimensional robust statistics. Specially, our contributions are the following:* We provide a general reduction from any robust estimation scheme to Bayesian approximately strategy-proof mechanism. This leads to new strategy-proof mechanisms for Gaussian and bounded moment distributions, by leveraging recent advances in robust statistics. * We show that the Lugosi-Mendelson median that arises from heavy-tailed statistics can be used to obtain Bayesian approximately strategy-proof single-facility mechanism with asymptotically optimal social cost, under mild distributional assumptions. * We provide Bayesian approximately strategy-proof multi-facility mechanisms for Gaussian mixture distributions with nearly optimal social cost. Finally, all of the mechanisms are computationally efficient, even in high-dimensions.