Approximates tr(log(M)) using tr(log(T)) where T is the projection of M into the krylov subspace K(M, v) where v is a sample from the metric M (i.E. v = v_lh + v_pr where v_lh/v_pr are samples from the likelihood/prior metric, respectively. In addition, the projected sample is constructed by taking v_pr projecting out the subspace K(M,v) using its eigen-basis. This ensures that both, the prior dominated part of v and the part already covered by tr(log(T)) is projected out.