  Standard Bayesian credible-region theory for constructing an error region on
the unique estimator of an unknown state in general quantum-state tomography to
calculate its size and credibility relies on heavy Monte~Carlo sampling of the
state space followed by sample rejection. This conventional method typically
gives negligible yield for very small error regions originating from large
datasets. We propose an operational reformulated theory to compute both size
and credibility from region-average quantities that in principle convey
information about behavior of these two properties as the credible-region
changes. We next suggest the accelerated hit-and-run Monte~Carlo sampling,
customized to the construction of Bayesian error-regions, to efficiently
compute region-average quantities, and provide its complexity estimates for
quantum states. Finally by understanding size as the region-average distance
between two states in the region (measured for instance with either the
Hilbert-Schmidt, trace-class or Bures distance), we derive approximation
formulas to analytically estimate both distance-induced size and credibility
under the pseudo-Bloch parametrization without resorting to any Monte~Carlo
computation.
