A power-prior posterior minimizing a weighted sum of Amari's alpha-divergences is exactly an alpha-geodesic between the no-borrowing and full-borrowing posteriors, with alpha controlling robustness.
(2020), ‘New insights and perspectives on the natural gradient method’,Journal of Machine Learning Research 21(146), 1–76
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Generalized Power Priors for Improved Bayesian Inference with Historical Data
A power-prior posterior minimizing a weighted sum of Amari's alpha-divergences is exactly an alpha-geodesic between the no-borrowing and full-borrowing posteriors, with alpha controlling robustness.