  The overall predictive uncertainty of a trained predictor can be decomposed
into separate contributions due to epistemic and aleatoric uncertainty. Under a
Bayesian formulation, assuming a well-specified model, the two contributions
can be exactly expressed (for the log-loss) or bounded (for more general
losses) in terms of information-theoretic quantities (Xu and Raginsky, 2020).
This paper addresses the study of epistemic uncertainty within an
information-theoretic framework in the broader setting of Bayesian
meta-learning. A general hierarchical Bayesian model is assumed in which
hyperparameters determine the per-task priors of the model parameters. Exact
characterizations (for the log-loss) and bounds (for more general losses) are
derived for the epistemic uncertainty -quantified by the minimum excess
meta-risk (MEMR)- of optimal meta-learning rules. This characterization is
leveraged to bring insights into the dependence of the epistemic uncertainty on
the number of tasks and on the amount of per-task training data. Experiments
are presented that use the proposed information-theoretic bounds, evaluated via
neural mutual information estimators, to compare the performance of
conventional learning and meta-learning as the number of meta-learning tasks
increases.
