Self-Consistent Stochastic Interpolants for Covariance Prior and Posterior Learning

Given many noisy covariance estimates, each computed from a few Gaussian samples with its own underlying covariance, how do we sample the covariance that produced a given estimate when no clean covariance is ever seen? This is an inverse problem: we also know the sampling mechanism, so each estimate is a draw from an unknown prior, seen through a channel we can simulate. Expectation–maximization over distributions undoes that blur, but every step requires sampling the posterior of the current prior, which no formula provides. We show that a conditional stochastic interpolant trained on its own reconstructions supplies that sampler and turns the iteration into an algorithm. Two versions target the same posterior over the whole matrix: one transports the matrix itself; the other, for a rotationally invariant prior, learns only the eigenvalue distribution and draws the relative orientation from an explicit conditional law, a completion that adds no distributional error. We prove identifiability exactly when each estimate uses at least as many samples as dimensions, and show that progress in what is observable is weaker than recovery of the prior, which needs further conditions. On synthetic data we recover both invariant and structured priors from noisy estimates alone, and on EEG and market records the posterior improves calibration and decisions.