  Statistical Inference is the process of determining a probability
distribution over the space of parameters of a model given a data set. As more
data becomes available this probability distribution becomes updated via the
application of Bayes' theorem. We present a treatment of this Bayesian updating
process as a continuous dynamical system. Statistical inference is then
governed by a first order differential equation describing a trajectory or flow
in the information geometry determined by a parametric family of models. We
solve this equation for some simple models and show that when the
Cram\'{e}r-Rao bound is saturated the learning rate is governed by a simple
$1/T$ power-law, with $T$ a time-like variable denoting the quantity of data.
The presence of hidden variables can be incorporated in this setting, leading
to an additional driving term in the resulting flow equation. We illustrate
this with both analytic and numerical examples based on Gaussians and Gaussian
Random Processes and inference of the coupling constant in the 1D Ising model.
Finally we compare the qualitative behaviour exhibited by Bayesian flows to the
training of various neural networks on benchmarked data sets such as MNIST and
CIFAR10 and show how that for networks exhibiting small final losses the simple
power-law is also satisfied.
