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Combining predictions from different models is a central problem in Bayesianinference and machine learning more broadly. Currently, these predictivedistributions are almost exclusively combined using linear mixtures such asBayesian model averaging, Bayesian stacking, and mixture of experts. Suchlinear mixtures impose idiosyncrasies that might be undesirable for someapplications, such as multi-modality. While there exist alternative strategies(e.g. geometric bridge or superposition), optimising their parameters usuallyinvolves computing an intractable normalising constant repeatedly. We presenttwo novel Bayesian model combination tools. These are generalisations of modelstacking, but combine posterior densities by log-linear pooling (locking) andquantum superposition (quacking). To optimise model weights while avoiding theburden of normalising constants, we investigate the Hyvarinen score of thecombined posterior predictions. We demonstrate locking with an illustrativeexample and discuss its practical application with importance sampling.

Locking and Quacking: Stacking Bayesian model predictions by log-pooling and superposition