Hence, the same simplification would apply in a larger joint probability expression such as the one in this model, composed of Dirichlet-multinomial densities plus factors for many other random variables dependent on the values of the categorical variables.
Consequently, the same reduction would be applicable to a more complex joint probability expression like the one presented in this framework, which combines Dirichlet-multinomial distributions with additional factors accounting for numerous other stochastic variables influenced by the categorical values.