Grouped Dirichlet distribution
In statistics, the grouped Dirichlet distribution (GDD) is a multivariate generalization of the Dirichlet distribution It was first described by Ng et al. 2008. The Grouped Dirichlet distribution arises in the analysis of categorical data where some observations could fall into any of a set of other 'crisp' category. For example, one may have a data set consisting of cases and controls under two different conditions. With complete data, the cross-classification of disease status forms a 2(case/control)-x-(condition/no-condition) table with cell probabilities If, however, the data includes, say, non-respondents which are known to  be controls or cases, then the cross-classification of disease status forms a 2-x-3 table. The probability of the last column is the sum of the probabilities of the first two columns in each row, e.g. The GDD allows the full estimation of the cell probabilities under such aggregation conditions. Probability Distribution. Consider the closed simplex set formula_1 and formula_2. Writing formula_3 for the first formula_4 elements of a member of formula_5, the distribution of formula_6 for two partitions has a density function given by where formula_8 is the multivariate beta function. Ng et al. went on to define  an "m" partition grouped Dirichlet distribution with density of formula_9 given by where formula_11 is a vector of integers with formula_12. The normalizing constant given by The authors went on to use these distributions in the context of three different applications in medical science.