  Using a Bayesian methodology, we introduce the maximum a posteriori~(MAP)
estimator for quantum state and process tomography. The maximum likelihood,
hedged maximum likelihood, maximum likelihood-maximum entropy estimator, and
estimators of this general type, can be viewed as special cases of the MAP
estimator. The MAP, like the Bayes mean estimator includes prior knowledge,
however for cases of interest to tomography can take advantage of convex
optimization tools making it numerically easier to compute. We show how the MAP
and other Bayesian quantum state estimators can be corrected for noise produced
if the quantum state passes through a noisy quantum channel prior to
measurement.
