The decentralization of decision-making is analyzed when agents may have information that is incomplete and possibly exclusive. Theorems provide conditions under which there exists a mechanism whose Bayesian equilibria coincide with a desired collection of social choice functions. The first theorem characterizes Bayesian implementation in economic environments with three or more individuals. The second theorem extends the analysis to noneconomic environments. An example exhibits differences between Bayesian implementation and Nash implementation. Copyright 1991 by The Econometric Society.