  Computing size and credibility of Bayesian credible regions for certifying
the reliability of any point estimator of an unknown parameter (such as a
quantum state, channel, phase, \emph{etc.}) relies on rejection sampling from
the entire parameter space that is practically infeasible for large datasets.
We reformulate the Bayesian credible-region theory to show that both properties
can be obtained solely from the average of log-likelihood over the region
itself, which is computable with direct region sampling. Neither rejection
sampling nor any geometrical knowledge about the whole parameter space is
necessary, so that general error certification now becomes feasible. We take
this region-average theory to the next level by generalizing size to the
average $l_p$-norm distance $(p>0)$ between a random region point and the
estimator, and present analytical formulas for $p=2$ to estimate
distance-induced size and credibility for any physical system and large
datasets, thus implying that asymptotic Bayesian error certification is
possible without any Monte~Carlo computation. All results are discussed in the
context of quantum-state tomography.
