Perfect aggregation of Bayesian analysis on compositional data

Sufficiency is a widely used concep t for reducing the dimensionality of a data set. Collecting data for a sufficient statistic is generally much easier and less expensive than collecting all of the available data. When the posterior distributions of a quantity of interest given the aggregate and disaggregate data are identical, perfect aggregation is said to hold, and in this case the aggregate data is a sufficient statistic for the quantity of interest. In this paper, the conditions for perfect aggregation are shown to depend on the functional form of the prior distribution. When the quantity of interest is the sum of some parameters in a vector having either a generalized Dirichlet or a Liouville distribution for analyzing compositional data, necessary and sufficient conditions for perfect aggregation are also established.