To my knowledge, Krylov subspace methods or classical iterative methods (say, Jacobi) are in general investigated in a "static" way; for instance, we consider the convergence of Jacobi method only by means of spectral radius which is somewhat "fixed". I wonder if there are any references on examining Krylov subspace methods from a statistical point of view.
I am uncertain about Krylov subspaces, but I know of stochastic algorithms that efficiently compute the SVD which is somehow related. Maybe this can give you a start for your investigation. There is, for example, the open source library RedSVD (https://code.google.com/p/redsvd/) with linked bibliography on its website.
Best regards
Sebastian
I note that there is a paper on this topic, see "(2012). An Efficient Bayesian Iterative Method for Linear Systems. Journal of Mathematical Research with Applications, Vol. 32 (No. 3), pp. 288–296." As you know, it is a Chinese journal. But I do not read it carefully. When I am available, I will read it again. I hope that it can be useful for you.
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