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 posterior probability that each data point belongs to a class. Alternatively, the mixing probabilities can be treated as free parameters, while assuming Gauss-Markov or more complex priors to regularize those mixing probabilities. However, these approaches are constrained by the shape of the prior and often lead to complicated or intractable inference. Here, we propose a new parametrization of the Dirichlet distribution to flexibly regularize the mixing probabilities of over-parametrized mixture distributions. Using the Expectation-Maximization approach -- to define any linear update rule for the mixing probabilities, including many smoothing regularizations as special cases.