  A novel method to propagate uncertainty through the soft-thresholding
nonlinearity is proposed in this paper. At every layer the current distribution
of the target vector is represented as a spike and slab distribution, which
represents the probabilities of each variable being zero, or
Gaussian-distributed. Using the proposed method of uncertainty propagation, the
gradients of the logarithms of normalisation constants are derived, that can be
used to update a weight distribution. A novel Bayesian neural network for
sparse coding is designed utilising both the proposed method of uncertainty
propagation and Bayesian inference algorithm.
