Implementation with Uncertain Evidence

We study a full implementation problem with a state unknown to the designer but known to agents, where agents have uncertain evidence privately drawn from state-dependent distributions. Stochastic evidence enables ``perfect deceptions,'' where agents' reports can mimic the evidence distribution of a false state, making differentiation impossible for any mechanism. This yields our main result: a necessary and sufficient condition, No Perfect Deceptions (NPD), for implementation in (mixed-strategy) Bayesian Nash equilibria. The solution requires novel techniques like belief elicitation via competing scoring rules, and an endogenous ``test allocation'' using the evidence structure. For informationally small agents (McLean and Postlewaite (2002)), a generalized condition (GNPD) is sufficient. Our mechanisms work for two or more agents, avoid integer/modulo games, and use limited liability transfers that vanish in equilibrium.