  We propose a feed-forward inference method applicable to belief and neural
networks. In a belief network, the method estimates an approximate factorized
posterior of all hidden units given the input. In neural networks the method
propagates uncertainty of the input through all the layers. In neural networks
with injected noise, the method analytically takes into account uncertainties
resulting from this noise. Such feed-forward analytic propagation is
differentiable in parameters and can be trained end-to-end. Compared to
standard NN, which can be viewed as propagating only the means, we propagate
the mean and variance. The method can be useful in all scenarios that require
knowledge of the neuron statistics, e.g. when dealing with uncertain inputs,
considering sigmoid activations as probabilities of Bernoulli units, training
the models regularized by injected noise (dropout) or estimating activation
statistics over the dataset (as needed for normalization methods). In the
experiments we show the possible utility of the method in all these tasks as
well as its current limitations.
