  A wide variety of battery models are available, and it is not always obvious
which model `best' describes a dataset. This paper presents a Bayesian model
selection approach using Bayesian quadrature. The model evidence is adopted as
the selection metric, choosing the simplest model that describes the data, in
the spirit of Occam's razor. However, estimating this requires integral
computations over parameter space, which is usually prohibitively expensive.
Bayesian quadrature offers sample-efficient integration via model-based
inference that minimises the number of battery model evaluations. The posterior
distribution of model parameters can also be inferred as a byproduct without
further computation. Here, the simplest lithium-ion battery models, equivalent
circuit models, were used to analyse the sensitivity of the selection criterion
to given different datasets and model configurations. We show that popular
model selection criteria, such as root-mean-square error and Bayesian
information criterion, can fail to select a parsimonious model in the case of a
multimodal posterior. The model evidence can spot the optimal model in such
cases, simultaneously providing the variance of the evidence inference itself
as an indication of confidence. We also show that Bayesian quadrature can
compute the evidence faster than popular Monte Carlo based solvers.
