We calibrate a stochastic computer simulation model of ‘moderate’ computational expense. The simulator is an imperfect representation of reality, and we recognize this discrepancy to ensure a reliable calibration. The calibration model combines a Gaussian process emulator of the likelihood surface with importance sampling. Changing the discrepancy specification changes only the importance weights, which lets us investigate sensitivity to different discrepancy specifications at little computational cost. We present a case study of a natural history model that has been used to characterize UK bowel cancer incidence. Datasets and computer code are provided as supplementary material.