  We outline a new method to compute the Bayes Factor for model selection which
bypasses the Bayesian Evidence. Our method combines multiple models into a
single, nested, Supermodel using one or more hyperparameters. Since the models
are now nested the Bayes Factors between the models can be efficiently computed
using the Savage-Dickey Density Ratio (SDDR). In this way model selection
becomes a problem of parameter estimation. We consider two ways of constructing
the supermodel in detail: one based on combined models, and a second based on
combined likelihoods. We report on these two approaches for a Gaussian linear
model for which the Bayesian evidence can be calculated analytically and a toy
nonlinear problem. Unlike the combined model approach, where a standard Monte
Carlo Markov Chain (MCMC) struggles, the combined-likelihood approach fares
much better in providing a reliable estimate of the log-Bayes Factor. This
scheme potentially opens the way to computationally efficient ways to compute
Bayes Factors in high dimensions that exploit the good scaling properties of
MCMC, as compared to methods such as nested sampling that fail for high
dimensions.
