  We study pointwise estimation and uncertainty quantification for a sparse
variational Gaussian process method with eigenvector inducing variables. For a
rescaled Brownian motion prior, we derive theoretical guarantees and
limitations for the frequentist size and coverage of pointwise credible sets.
For sufficiently many inducing variables, we precisely characterize the
asymptotic frequentist coverage, deducing when credible sets from this
variational method are conservative and when overconfident/misleading. We
numerically illustrate the applicability of our results and discuss connections
with other common Gaussian process priors.
