  As machine learning models are increasingly deployed in dynamic environments,
it becomes paramount to assess and quantify uncertainties associated with
distribution shifts. A distribution shift occurs when the underlying
data-generating process changes, leading to a deviation in the model's
performance. The prediction interval, which captures the range of likely
outcomes for a given prediction, serves as a crucial tool for characterizing
uncertainties induced by their underlying distribution. In this paper, we
propose methodologies for aggregating prediction intervals to obtain one with
minimal width and adequate coverage on the target domain under unsupervised
domain shift, under which we have labeled samples from a related source domain
and unlabeled covariates from the target domain. Our analysis encompasses
scenarios where the source and the target domain are related via i) a bounded
density ratio, and ii) a measure-preserving transformation. Our proposed
methodologies are computationally efficient and easy to implement. Beyond
illustrating the performance of our method through a real-world dataset, we
also delve into the theoretical details. This includes establishing rigorous
theoretical guarantees, coupled with finite sample bounds, regarding the
coverage and width of our prediction intervals. Our approach excels in
practical applications and is underpinned by a solid theoretical framework,
ensuring its reliability and effectiveness across diverse contexts.
