Motivated by the likelihood functions of incomplete categorical data in different background, we have proposed and developed in a serious of papers with collaborators two new families of distributions as new tools. They are Grouped Dirichlet Distribution and Nested Dirichlet Distribution, both encompassing the classical Dirichlet distribution as a special case. In this talk, we shall separately review the properties of these two families of distributions in their own rights and demonstrate with real and simulated data how these two expanded families of distributions emerge as new tools for both maximum likelihood and Bayesian inference of incomplete categorical data. The advantages of using these new tools will be discussed in detail.