  Bayesian evidence is a standard method used for comparing the ability of
different models to fit available data and is used extensively in cosmology.
However, since the evidence calculation involves performing an integral of the
likelihood function over the entire space of model parameters this can be
prohibitively expensive in terms of both CPU and time consumption. For example,
in the simplest $\Lambda$CDM model and using CMB data from the Planck
satellite, the dimensionality of the model space is over 30 (typically 6
cosmological parameters and 28 nuisance parameters). Even the simplest possible
model requires $\mathcal{O}(10^6)$ calls to an Einstein--Boltzmann solver such
as CLASS or CAMB and takes several days.
  Here we present calculations of Bayesian evidence using the CONNECT framework
to calculate cosmological observables. We demonstrate that we can achieve
results comparable to those obtained using Einstein--Boltzmann solvers, but at
a minute fraction of the computational cost. As a test case, we then go on to
compute Bayesian evidence ratios for a selection of slow-roll inflationary
models.
  In the setup presented here, the total computation time is completely
dominated by the likelihood function calculation which now becomes the main
bottleneck for increasing computation speed.
