Matrix variate Dirichlet distribution
In statistics, the matrix variate Dirichlet distribution is a generalization of the matrix variate beta distribution and of the Dirichlet distribution. Suppose [math]\displaystyle{ U_1,\ldots,U_r }[/math] are [math]\displaystyle{ p\times p }[/math] positive definite matrices with [math]\displaystyle{ I_p-\sum_{i=1}^rU_i }[/math] also positive-definite, where [math]\displaystyle{ I_p }[/math] is the [math]\displaystyle{ p\times p }[/math] identity matrix. Then we say that the [math]\displaystyle{ U_i }[/math] have a matrix variate Dirichlet distribution, [math]\displaystyle{ \left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_r;a_{r+1}\right) }[/math], if their joint probability density function is
- [math]\displaystyle{ \left\{\beta_p\left(a_1,\ldots,a_r,a_{r+1}\right)\right\}^{-1}\prod_{i=1}^{r}\det\left(U_i\right)^{a_i-(p+1)/2}\det\left(I_p-\sum_{i=1}^rU_i\right)^{a_{r+1}-(p+1)/2} }[/math]
where [math]\displaystyle{ a_i\gt (p-1)/2,i=1,\ldots,r+1 }[/math] and [math]\displaystyle{ \beta_p\left(\cdots\right) }[/math] is the multivariate beta function.
If we write [math]\displaystyle{ U_{r+1}=I_p-\sum_{i=1}^r U_i }[/math] then the PDF takes the simpler form
- [math]\displaystyle{ \left\{\beta_p\left(a_1,\ldots,a_{r+1}\right)\right\}^{-1}\prod_{i=1}^{r+1}\det\left(U_i\right)^{a_i-(p+1)/2}, }[/math]
on the understanding that [math]\displaystyle{ \sum_{i=1}^{r+1}U_i=I_p }[/math].
Theorems
generalization of chi square-Dirichlet result
Suppose [math]\displaystyle{ S_i\sim W_p\left(n_i,\Sigma\right),i=1,\ldots,r+1 }[/math] are independently distributed Wishart [math]\displaystyle{ p\times p }[/math] positive definite matrices. Then, defining [math]\displaystyle{ U_i=S^{-1/2}S_i\left(S^{-1/2}\right)^T }[/math] (where [math]\displaystyle{ S=\sum_{i=1}^{r+1}S_i }[/math] is the sum of the matrices and [math]\displaystyle{ S^{1/2}\left(S^{-1/2}\right)^T }[/math] is any reasonable factorization of [math]\displaystyle{ S }[/math]), we have
- [math]\displaystyle{ \left(U_1,\ldots,U_r\right)\sim D_p\left(n_1/2,...,n_{r+1}/2\right). }[/math]
Marginal distribution
If [math]\displaystyle{ \left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_{r+1}\right) }[/math], and if [math]\displaystyle{ s\leq r }[/math], then:
- [math]\displaystyle{ \left(U_1,\ldots,U_s\right)\sim D_p\left(a_1,\ldots,a_s,\sum_{i=s+1}^{r+1}a_i\right) }[/math]
Conditional distribution
Also, with the same notation as above, the density of [math]\displaystyle{ \left(U_{s+1},\ldots,U_r\right)\left|\left(U_1,\ldots,U_s\right)\right. }[/math] is given by
- [math]\displaystyle{ \frac{ \prod_{i=s+1}^{r+1}\det\left(U_i\right)^{a_i-(p+1)/2} }{ \beta_p\left(a_{s+1},\ldots,a_{r+1}\right)\det\left(I_p-\sum_{i=1}^{s}U_i\right)^{\sum_{i=s+1}^{r+1}a_i-(p+1)/2} } }[/math]
where we write [math]\displaystyle{ U_{r+1} = I_p-\sum_{i=1}^rU_i }[/math].
partitioned distribution
Suppose [math]\displaystyle{ \left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_{r+1}\right) }[/math] and suppose that [math]\displaystyle{ S_1,\ldots,S_t }[/math] is a partition of [math]\displaystyle{ \left[r+1\right]=\left\{1,\ldots r+1\right\} }[/math] (that is, [math]\displaystyle{ \cup_{i=1}^tS_i=\left[r+1\right] }[/math] and [math]\displaystyle{ S_i\cap S_j=\emptyset }[/math] if [math]\displaystyle{ i\neq j }[/math]). Then, writing [math]\displaystyle{ U_{(j)}=\sum_{i\in S_j}U_i }[/math] and [math]\displaystyle{ a_{(j)}=\sum_{i\in S_j}a_i }[/math] (with [math]\displaystyle{ U_{r+1}=I_p-\sum_{i=1}^r U_r }[/math]), we have:
- [math]\displaystyle{ \left(U_{(1)},\ldots U_{(t)}\right)\sim D_p\left(a_{(1)},\ldots,a_{(t)}\right). }[/math]
partitions
Suppose [math]\displaystyle{ \left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_{r+1}\right) }[/math]. Define
- [math]\displaystyle{ U_i= \left( \begin{array}{rr} U_{11(i)} & U_{12(i)} \\ U_{21(i)} & U_{22(i)} \end{array} \right) \qquad i=1,\ldots,r }[/math]
where [math]\displaystyle{ U_{11(i)} }[/math] is [math]\displaystyle{ p_1\times p_1 }[/math] and [math]\displaystyle{ U_{22(i)} }[/math] is [math]\displaystyle{ p_2\times p_2 }[/math]. Writing the Schur complement [math]\displaystyle{ U_{22\cdot 1(i)} = U_{21(i)} U_{11(i)}^{-1}U_{12(i)} }[/math] we have
- [math]\displaystyle{ \left(U_{11(1)},\ldots,U_{11(r)}\right)\sim D_{p_1}\left(a_1,\ldots,a_{r+1}\right) }[/math]
and
- [math]\displaystyle{ \left(U_{22.1(1)},\ldots,U_{22.1(r)}\right)\sim D_{p_2}\left(a_1-p_1/2,\ldots,a_r-p_1/2,a_{r+1}-p_1/2+p_1r/2\right). }[/math]
See also
References
A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.
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