  Calculation of Bayesian posteriors and model evidences typically requires
numerical integration. Bayesian quadrature (BQ), a surrogate-model-based
approach to numerical integration, is capable of superb sample efficiency, but
its lack of parallelisation has hindered its practical applications. In this
work, we propose a parallelised (batch) BQ method, employing techniques from
kernel quadrature, that possesses an empirically exponential convergence rate.
Additionally, just as with Nested Sampling, our method permits simultaneous
inference of both posteriors and model evidence. Samples from our BQ surrogate
model are re-selected to give a sparse set of samples, via a kernel
recombination algorithm, requiring negligible additional time to increase the
batch size. Empirically, we find that our approach significantly outperforms
the sampling efficiency of both state-of-the-art BQ techniques and Nested
Sampling in various real-world datasets, including lithium-ion battery
analytics.
