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.{{cite journal| last=Ng| first=Kai Wang| title=Grouped Dirichlet distribution: A new tool for incomplete categorical data analysis| journal=Journal of Multivariate Analysis| year=2008| volume=99| pages=490–509}} 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

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	Treatment	No Treatment
Controls	θ₁	θ₂
Cases	θ₃	θ₄

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.

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	Treatment	No Treatment	Missing
Controls	θ₁	θ₂	θ₁+θ₂
Cases	θ₃	θ₄	θ₃+θ₄

The GDD allows the full estimation of the cell probabilities under such aggregation conditions.

Probability Distribution

Consider the closed simplex set $\mathcal{T}_n=\left\{\left(x_1,\ldots x_n\right)\left|x_i\geq 0, i=1,\cdots,n, \sum_{i=1}^n x_n =1\right.\right\}$ and

$\mathbf{x}\in\mathcal{T}_n$ . Writing $\mathbf{x}_{-n}=\left(x_1,\ldots,x_{n-1}\right)$ for the first $n-1$ elements of a member of $\mathcal{T}_n$ , the distribution of $\mathbf{x}$ for two partitions has a density function given by

: $\operatorname{GD}_{n,2,s}\left(\left.\mathbf{x}_{-n}\right|\mathbf{a},\mathbf{b}\right)=
\frac{
\left(\prod_{i=1} ^n x_i^{a_i-1}\right)\cdot
\left(\sum_{i=1} ^s x_i \right)^{b_1}\cdot
\left(\sum_{i=s+1} ^n x_i \right)^{b_2}
}{
\operatorname{\Beta}\left(a_1,\ldots,a_s\right)\cdot
\operatorname{\Beta}\left(a_{s+1},\ldots,a_n\right)\cdot
\operatorname{\Beta}\left(b_1+\sum_{i=1}^sa_i,b_2+\sum_{i=s+1}^n a_i\right)
}$

where $\operatorname{\Beta}\left(\mathbf{a}\right)$ is the Multivariate beta function.

Ng et al. went on to define an m partition grouped Dirichlet distribution with density of $\mathbf{x}_{-n}$ given by

: $\operatorname{GD}_{n,m,\mathbf{s}}\left(\left.\mathbf{x}_{-n}\right|\mathbf{a},\mathbf{b}\right) =
c_m^{-1}\cdot
\left(\prod_{i=1}^n x_i^{a_i-1}\right)\cdot
\prod_{j=1}^m\left(\sum_{k=s_{j-1}+1}^{s_j}x_k\right)^{b_j}$

where $\mathbf{s} = \left(s_1,\ldots,s_m\right)$ is a vector of integers with $0=s_0 . The normalizing constant given by$

: $c_m=\left\{\prod_{j=1}^m\operatorname{\Beta}\left(a_{s_{j-1}+1},\ldots,a_{s_j}\right)\right\}\cdot
\operatorname{\Beta}\left(b_1+\sum_{k=1}^{s_1}a_k,\ldots,b_m+\sum_{k=s_{m-1}+1}^{s_m}a_k\right)$

The authors went on to use these distributions in the context of three different applications in medical science.

References

Category:Multivariate continuous distributions

Category:Conjugate prior distributions

Category:Exponential family distributions

Category:Continuous distributions