Probing Bayesian credible regions intrinsically: a feasible error certification for physical systems

Abstract: 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.