  Classical Krylov subspace projection methods for the solution of linear
problem $Ax = b$ output an approximate solution $\widetilde{x}\simeq x$.
Recently, it has been recognized that projection methods can be understood from
a statistical perspective. These probabilistic projection methods return a
distribution $p(\widetilde{x})$ in place of a point estimate $\widetilde{x}$.
The resulting uncertainty, codified as a distribution, can, in theory, be
meaningfully combined with other uncertainties, can be propagated through
computational pipelines, and can be used in the framework of probabilistic
decision theory. The problem we address is that the current probabilistic
projection methods lead to the poorly calibrated posterior distribution. We
improve the covariance matrix from previous works in a way that it does not
contain such undesirable objects as $A^{-1}$ or $A^{-1}A^{-T}$, results in
nontrivial uncertainty, and reproduces an arbitrary projection method as a mean
of the posterior distribution. We also propose a variant that is numerically
inexpensive in the case the uncertainty is calibrated a priori. Since it
usually is not, we put forward a practical way to calibrate uncertainty that
performs reasonably well, albeit at the expense of roughly doubling the
numerical cost of the underlying projection method.
