  Overparameterised deep neural networks (DNNs) are highly expressive and so
can, in principle, generate almost any function that fits a training dataset
with zero error. The vast majority of these functions will perform poorly on
unseen data, and yet in practice DNNs often generalise remarkably well. This
success suggests that a trained DNN must have a strong inductive bias towards
functions with low generalisation error. Here we empirically investigate this
inductive bias by calculating, for a range of architectures and datasets, the
probability $P_{SGD}(f\mid S)$ that an overparameterised DNN, trained with
stochastic gradient descent (SGD) or one of its variants, converges on a
function $f$ consistent with a training set $S$. We also use Gaussian processes
to estimate the Bayesian posterior probability $P_B(f\mid S)$ that the DNN
expresses $f$ upon random sampling of its parameters, conditioned on $S$.
  Our main findings are that $P_{SGD}(f\mid S)$ correlates remarkably well with
$P_B(f\mid S)$ and that $P_B(f\mid S)$ is strongly biased towards low-error and
low complexity functions. These results imply that strong inductive bias in the
parameter-function map (which determines $P_B(f\mid S)$), rather than a special
property of SGD, is the primary explanation for why DNNs generalise so well in
the overparameterised regime.
  While our results suggest that the Bayesian posterior $P_B(f\mid S)$ is the
first order determinant of $P_{SGD}(f\mid S)$, there remain second order
differences that are sensitive to hyperparameter tuning. A function probability
picture, based on $P_{SGD}(f\mid S)$ and/or $P_B(f\mid S)$, can shed new light
on the way that variations in architecture or hyperparameter settings such as
batch size, learning rate, and optimiser choice, affect DNN performance.
