  To represent the biological variability of clinical neuroimaging populations,
it is vital to be able to combine data across scanners and studies. However,
different MRI scanners produce images with different characteristics, resulting
in a domain shift known as the `harmonisation problem'. Additionally,
neuroimaging data is inherently personal in nature, leading to data privacy
concerns when sharing the data. To overcome these barriers, we propose an
Unsupervised Source-Free Domain Adaptation (SFDA) method, SFHarmony. Through
modelling the imaging features as a Gaussian Mixture Model and minimising an
adapted Bhattacharyya distance between the source and target features, we can
create a model that performs well for the target data whilst having a shared
feature representation across the data domains, without needing access to the
source data for adaptation or target labels. We demonstrate the performance of
our method on simulated and real domain shifts, showing that the approach is
applicable to classification, segmentation and regression tasks, requiring no
changes to the algorithm. Our method outperforms existing SFDA approaches
across a range of realistic data scenarios, demonstrating the potential utility
of our approach for MRI harmonisation and general SFDA problems. Our code is
available at \url{https://github.com/nkdinsdale/SFHarmony}.
