  Evidence Networks can enable Bayesian model comparison when state-of-the-art
methods (e.g. nested sampling) fail and even when likelihoods or priors are
intractable or unknown. Bayesian model comparison, i.e. the computation of
Bayes factors or evidence ratios, can be cast as an optimization problem.
Though the Bayesian interpretation of optimal classification is well-known,
here we change perspective and present classes of loss functions that result in
fast, amortized neural estimators that directly estimate convenient functions
of the Bayes factor. This mitigates numerical inaccuracies associated with
estimating individual model probabilities. We introduce the leaky parity-odd
power (l-POP) transform, leading to the novel ``l-POP-Exponential'' loss
function. We explore neural density estimation for data probability in
different models, showing it to be less accurate and scalable than Evidence
Networks. Multiple real-world and synthetic examples illustrate that Evidence
Networks are explicitly independent of dimensionality of the parameter space
and scale mildly with the complexity of the posterior probability density
function. This simple yet powerful approach has broad implications for model
inference tasks. As an application of Evidence Networks to real-world data we
compute the Bayes factor for two models with gravitational lensing data of the
Dark Energy Survey. We briefly discuss applications of our methods to other,
related problems of model comparison and evaluation in implicit inference
settings.
