  We consider the problem of Bayesian regression with trustworthy uncertainty
quantification. We define that the uncertainty quantification is trustworthy if
the ground truth can be captured by intervals dependent on the predictive
distributions with a pre-specified probability. Furthermore, we propose,
Trust-Bayes, a novel optimization framework for Bayesian meta learning which is
cognizant of trustworthy uncertainty quantification without explicit
assumptions on the prior model/distribution of the functions. We characterize
the lower bounds of the probabilities of the ground truth being captured by the
specified intervals and analyze the sample complexity with respect to the
feasible probability for trustworthy uncertainty quantification. Monte Carlo
simulation of a case study using Gaussian process regression is conducted for
verification and comparison with the Meta-prior algorithm.
