#### Compromise-free Bayesian neural networks

##### Kamran Javid, Will Handley, Mike Hobson, Anthony Lasenby

We conduct a thorough analysis of the relationship between the out-of-sample performance and the Bayesian evidence of Bayesian neural networks (BNNs) using the Boston housing dataset, as well as looking at the performance of ensembles of BNNs. We numerically sample without compromise the full network posterior and obtain estimates of the Bayesian evidence using the publicly available version of the nested sampling algorithm \PolyChord{}\footnote[6]{\url{https://github.com/PolyChord/PolyChordLite}} \citep{handley2015polychord1, handley2015polychord2}, considering network models with up to $156$ trainable parameters\footnote[5]{Our BNN implementation is available at \url{https://github.com/SuperKam91/bnn}}\citep{kamran_javid_2020_3758553}. The networks have between zero and four hidden layers, either $\tanh$ or ReLU activation functions, and with and without hierarchical priors \citep{mackay1992practical, neal2012bayesian}. The ensembles of BNNs are obtained by determining the posterior distribution over networks, from the posterior samples of individual BNNs re-weighted by the associated Bayesian evidence values. From the out-of-sample performance of the BNNs with ReLU activations, it is clear that they outperform BNNs of the same architecture with $\tanh$ activations, and evidence values corresponding to the former reflect this in their relatively high values. Looking at the models with hierarchical priors, there is a good correlation between out-of-sample performance and evidence, as was found in \cite{mackay1992practical}, as well as a remarkable symmetry between the evidence versus model size and out-of-sample performance versus model size planes. The BNNs predictively outperform the equivalent neural networks trained with a traditional backpropagation approach, and Bayesian marginalising/ensembling over architectures acts to further improve performance.

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