Implicitly Bayesian Prediction Rules in Deep Learning

The Bayesian approach leads to coherent updates of predictions under new data, which makes adhering to Bayesian principles appealing in decision-making contexts. Traditionally, integrating Bayesian principles into models like deep neural networks involves setting priors on parameters and approximating posteriors. This is done despite the fact that, typically, priors on parameters reflect any prior beliefs only insofar as they dictate function space behaviour. In this paper, we rethink this approach and consider what properties characterise a prediction rule as being Bayesian. Algorithms meeting such criteria can be deemed implicitly Bayesian - they make the same predictions as some Bayesian model, without explicitly manifesting priors and posteriors. We argue this might be a more fruitful approach towards integrating Bayesian principles into deep learning. In this paper, we propose how to measure how close a general prediction rule is to being implicitly Bayesian, and empirically evaluate multiple prediction strategies using our approach. We also show theoretically that agents relying on non-implicitly Bayesian prediction rules can be easily exploited in adversarial betting settings.