Pigouvian taxes are typically imposed in situations where there is imperfect knowledge on the extent of damage caused by a producing firm. A regulator imposing imperfectly informed Pigouvian taxes may cause the firms that should (should not) produce to shut down (produce). In this paper we use a Bayesian information framework to identify optimal signal-conditioned taxes in the presence of such losses. The tax system involves reducing (increasing) taxes on firms identified as causing high (low) damage. Unfortunately, when an abatement decision has to be made, the tax system that minimizes production distortions also dampens the incentive to abate. In the absence of wrong-firm concerns, a regulator can solve the problem by not adjusting taxes for signal noise. When wrong-firm losses are a concern, the regulator has to trade off losses from distorted production incentives with losses from distorted abatement incentives. The most appropriate policy may involve a combination of instruments.