Before revision:  First, we show that a Dirichlet prior with an appropriate choice of parameters allows -- using the  Expectation-Maximization approach -- to define any linear update rule for the mixing probabilities, including many smoothing regularizations as special cases. Second, we demonstrate how to use this flexible design of the update rule to propagate segmentation information across layers of a deep network, and to train mixtures jointly across layers.
After revision:  First, we show that a Dirichlet prior with an appropriate choice of parameters allows -- using the Expectation-Maximization approach -- to define any linear update rule for the mixing probabilities, including spatial smoothing regularization as a special case. We then show that this flexible design of the update rule to propagate segmentation information across layers of a deep network, and to train mixtures jointly across layers.