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We generalize the Metropolis et al. random walk algorithm to the situationwhere the energy is noisy and can only be estimated. Two possible applicationsare for long range potentials and for mixed quantum-classical simulations. Ifthe noise is normally distributed we are able to modify the acceptanceprobability by applying a penalty to the energy difference and thereby achieveexact sampling even with very strong noise. When one has to estimate thevariance we have an approximate formula, good in the limit of large number ofindependent estimates. We argue that the penalty method is nearly optimal. Wealso adapt an existing method by Kennedy and Kuti and compare to the penaltymethod on a one dimensional double well.Comment: 17 pages, 7 figures, accepted to Journal of Chemical Physics;  Corrected swap of Figures 2 and 3. Added Figure

The penalty method for random walks with uncertain energies