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Modern neural networks have proven to be powerful function approximators, providing state-of-the-art performance in a multitude of applications. They however fall short in their ability to quantify confidence in their predictions - this is crucial in high-stakes applications that involve critical decision-making. Bayesian neural networks (BNNs) aim at solving this problem by placing a prior distribution over the network's parameters, thereby inducing a posterior distribution that encapsulates predictive uncertainty. While existing variants of BNNs based on Monte Carlo dropout produce reliable (albeit approximate) uncertainty estimates over in-distribution data, they tend to exhibit over-confidence in predictions made on target data whose feature distribution differs from the training data, i.e., the covariate shift setup. In this paper, we develop an approximate Bayesian inference scheme based on posterior regularisation, wherein unlabelled target data are used as "pseudo-labels" of model confidence that are used to regularise the model's loss on labelled source data. We show that this approach significantly improves the accuracy of uncertainty quantification on covariate-shifted data sets, with minimal modification to the underlying model architecture. We demonstrate the utility of our method in the context of transferring prognostic models of prostate cancer across globally diverse populations.