  This manuscript proposes a probabilistic framework for algorithms that
iteratively solve unconstrained linear problems $Bx = b$ with positive definite
$B$ for $x$. The goal is to replace the point estimates returned by existing
methods with a Gaussian posterior belief over the elements of the inverse of
$B$, which can be used to estimate errors. Recent probabilistic interpretations
of the secant family of quasi-Newton optimization algorithms are extended.
Combined with properties of the conjugate gradient algorithm, this leads to
uncertainty-calibrated methods with very limited cost overhead over conjugate
gradients, a self-contained novel interpretation of the quasi-Newton and
conjugate gradient algorithms, and a foundation for new nonlinear optimization
methods.
