Before revision:  We introduce a possibly singular Krylov prior covariance, under which the BayesCG posterior means coincide with the CG iterates and the posteriors can be computed efficiently. Because of its factored form, the Krylov prior is amenable to low-rank approximation, which produces an efficient BayesCG implementation as a CG method.
After revision:  We introduce a possibly singular Krylov prior covariance, under which the BayesCG posterior means coincide with the CG iterates and posterior covariances that can be propagated to subsequent computations. The covariances are low-rank approximation, which produces an efficient BayesCG implementation as a CG method.