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We study pointwise estimation and uncertainty quantification for a sparsevariational Gaussian process method with eigenvector inducing variables. For arescaled Brownian motion prior, we derive theoretical guarantees andlimitations for the frequentist size and coverage of pointwise credible sets.For sufficiently many inducing variables, we precisely characterize theasymptotic frequentist coverage, deducing when credible sets from thisvariational method are conservative and when overconfident/misleading. Wenumerically illustrate the applicability of our results and discuss connectionswith other common Gaussian process priors.

Pointwise uncertainty quantification for sparse variational Gaussian process regression with a Brownian motion prior