  There is growing interest in Bayesian clinical trial designs with informative
prior distributions, e.g. for extrapolation of adult data to pediatrics, or use
of external controls. While the classical type I error is commonly used to
evaluate such designs, it cannot be strictly controlled and it is acknowledged
that other metrics may be more appropriate. We focus on two common situations -
borrowing control data or information on the treatment contrast - and discuss
several fully probabilistic metrics to evaluate the risk of false positive
conclusions. Each metric requires specification of a design prior, which can
differ from the analysis prior and permits understanding of the behaviour of a
Bayesian design under scenarios where the analysis prior differs from the true
data generation process. The metrics include the average type I error and the
pre-posterior probability of a false positive result. We show that, when
borrowing control data, the average type I error is asymptotically (in certain
cases strictly) controlled when the analysis and design prior coincide. We
illustrate use of these Bayesian metrics with real applications, and discuss
how they could facilitate discussions between sponsors, regulators and other
stakeholders about the appropriateness of Bayesian borrowing designs for
pivotal studies.
