I have been learning variational bayesian mixture model. Appearantly, beta / dirichlet distribution are used as prior distribution. Why use these two distribution as priori and what is the principle behind this?
A prior distribution represents the prior believe concerning how likely different values of the parameter are. So at its core, a beta or Dirichlet ( [pedantic mode on] called after Johann Dirichlet, so writen with a capital letter [pedantic mode off] ) distribution is chosen by someone because that person thinks that it represents her or his believes. If your believes differ than you choose a different distribution.
However, not all distributions are equally sensible. The mixture parameter is a proportion that must remain between 0 and 1, so a normal distribution would not be the most sensible prior distribution for that parameter because that would assign a positive probility to impossible values (less than 0 or more than 1). A beta distirbution is constrained between 0 and 1 and fairly flexible, so you can represent a lot of different believes with that one distribution. This makes it a convenient choice. A Dirichlet distribution also ensures that the parameters add up to 1, which is typically required, though its correlation structure is somewhat restrictive. In that case it is a convenient choice.