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We provide a characterization of virtual Bayesian implementation in pure strategies for environments satisfying no-total-indifference. A social choice function in such environments is virtually Bayesian implementable if and only if it satisfies incentive compatibility and a condition we term virtual monotonicity. The latter is weaker than Bayesian monotonicity -known to be necessary for Bayesian implementation. Virtual monotonicity is weak in the sense that it is generically satisfied indoi:10.2139/ssrn.310425 fatcat:4uqp2gfugbdybed35pbdvqlrba