  Standard regularized training procedures correspond to maximizing a posterior
distribution over parameters, known as maximum a posteriori (MAP) estimation.
However, model parameters are of interest only insomuch as they combine with
the functional form of a model to provide a function that can make good
predictions. Moreover, the most likely parameters under the parameter posterior
do not generally correspond to the most likely function induced by the
parameter posterior. In fact, we can re-parametrize a model such that any
setting of parameters can maximize the parameter posterior. As an alternative,
we investigate the benefits and drawbacks of directly estimating the most
likely function implied by the model and the data. We show that this procedure
leads to pathological solutions when using neural networks and prove conditions
under which the procedure is well-behaved, as well as a scalable approximation.
Under these conditions, we find that function-space MAP estimation can lead to
flatter minima, better generalization, and improved robustness to overfitting.
