  Recent work has argued that stochastic gradient descent can approximate the
Bayesian uncertainty in model parameters near local minima. In this work we
develop a similar correspondence for minibatch natural gradient descent (NGD).
We prove that for sufficiently small learning rates, if the model predictions
on the training set approach the true conditional distribution of labels given
inputs, the stationary distribution of minibatch NGD approaches a Bayesian
posterior near local minima. The temperature $T = \epsilon N / (2B)$ is
controlled by the learning rate $\epsilon$, training set size $N$ and batch
size $B$. However minibatch NGD is not parameterisation invariant and it does
not sample a valid posterior away from local minima. We therefore propose a
novel optimiser, "stochastic NGD", which introduces the additional correction
terms required to preserve both properties.
