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In this paper we introduce the Minimum Phone Error (MPE) and Minimum Word Error (MWE) criteria for the discriminative training of HMM systems. The MPE/MWE criteria are smoothed approximations to the phone or word error rate respectively. We also discuss I-smoothing which is a novel technique for smoothing discriminative training criteria using statistics for maximum likelihood estimation (MLE). Experiments have been performed on the Switchboard/Call Home corpora of telephone conversations with up to 265 hours of training data. It is shown that for the maximum mutual information estimation (MMIE) criterion, I-smoothing reduces the word error rate (WER) by 0.4% absolute over the MMIE baseline. The combination of MPE and I-smoothing gives an improvement of 1 % over MMIE and a total reduction in WER of 4.8% absolute over the original MLE system.