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.
After revision:  First, we show that a Dirichlet prior with an appropriate choice of parameters allows -- using the Expectation-Maximization algorithm, we show that our approach allows us to define any linear update rule for the mixing probabilities, including many smoothing regularizations as special cases.