Convergence diagnostics are widely used to determine how many initial “burn-in” iterations should be discarded from the output of a Markov chain Monte Carlo (MCMC) sampler in the hope that the remaining samples are representative of the target distribution of interest. This paper demonstrates that some ways of applying convergence diagnostics may actually introduce bias into estimation based on the sampler output. To avoid this possibility, we recommend choosing the number of burn-in iterations r by applying convergence diagnostics to one or more pilot chains, and then basing estimation and inference on a separate long chain from which the first r iterations have been discarded.