In applications of Gaussian processes where quantification of uncertainty is of primary interest, it is necessary to accurately characterize the posterior distribution over covariance parameters. This paper proposes stochastic gradient-based inference to draw samples from the posterior distribution over covariance parameters with negligible bias and without the need to compute the marginal likelihood. In Gaussian process regression, this has the enormous advantage that stochastic gradients can be computed by solving linear systems only. A novel unbiased linear systems solver based on parallelizable covariance matrix-vector products is developed to accelerate the unbiased estimation of gradients. The results demonstrate the possibility to enable scalable and exact (in a Monte Carlo sense) quantification of uncertainty in Gaussian processes without imposing any special structures on the covariance or reducing the number of input vectors.
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