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We analyse the calibration of BayesCG under the Krylov prior, a probabilisticnumeric extension of the Conjugate Gradient (CG) method for solving systems oflinear equations with symmetric positive definite coefficient matrix.Calibration refers to the statistical quality of the posterior covariancesproduced by a solver. Since BayesCG is not calibrated in the strict existingnotion, we propose instead two test statistics that are necessary but notsufficient for calibration: the Z-statistic and the new S-statistic. We showanalytically and experimentally that under low-rank approximate Krylovposteriors, BayesCG exhibits desirable properties of a calibrated solver, isonly slightly optimistic, and is computationally competitive with CG.

Statistical Properties of the Probabilistic Numeric Linear Solver BayesCG