Matrix variate Dirichlet distribution explained

In statistics, the matrix variate Dirichlet distribution is a generalization of the matrix variate beta distribution and of the Dirichlet distribution.

Suppose

U _{1,\ldots,U} _{r} are p x p positive definite matrices with I _{p-\sum} ^{rU} ^{i} also positive-definite, where I _{p} is the p x p identity matrix. Then we say that the U _{i} have a matrix variate Dirichlet distribution, \left(U _{1,\ldots,U} _{r\right)\sim} D _{p\left(a} _{1,\ldots,a} _{r;a} _{r+1} \right) , if their joint probability density function is\left\{\beta _{p\left(a} _{1,\ldots,a} _{r,a} _{r+1} \right)\right\} ^{-1} \prod ^{r} _{i =1} \det\left(U ^{a _{i-(p+1)/2}} _{i\right)} \det\left(I _{p-\sum} ^{a _{r+1} - (p +1 )/ 2} ^{i\right)}

where

a _{i>(p-1)/2,i=1,\ldots,r+1} and \beta _{p\left( … \right)} is the multivariate beta function.

If we write

U _{r+1} = I _{p-\sum} ^{r} ^{i =1} U _{i} then the PDF takes the simpler form\left\{\beta _{p\left(a} _{1,\ldots,a} _{r+1} \right)\right\} ^{-1} \prod ^{r +1} _{i =1} \det\left(U ^{a _{i-(p+1)/2}} _{i\right)} ,

on the understanding that

\sum ^{r +1} _{i =1} U _{i=I} _{p} .

Theorems

generalization of chi square-Dirichlet result

Suppose

S _{i\sim} W _{p\left(n} _{i,\Sigma\right),i=1,\ldots,r+1} are independently distributed Wishart p x p positive definite matrices. Then, defining U ^{-1 / 2} _{i = S} S ^{-1 / 2} _{i\left(S} \right) ^{T} (where $S=\backslash sum\_^S\_i$ is the sum of the matrices and S ^{1/2} \left(S ^{-1/2} \right) ^{T} is any reasonable factorization of S ), we have\left(U _{1,\ldots,U} _{r\right)\sim} D _{p\left(n} _{1/2,...,n} _{r+1} / 2 \right).

Marginal distribution

If

\left(U _{1,\ldots,U} _{r\right)\sim} D _{p\left(a} _{1,\ldots,a} _{r+1} \right) , and if s\leq r , then:\left(U _{1,\ldots,U} _{s\right)\sim} D _{p\left(a} _{1,\ldots,a} _{s,\sum} ^{r +1} ^{i = s +1} a _{i\right)}

Conditional distribution

Also, with the same notation as above, the density of

\left(U _{s+1} ,\ldots,U _{r\right)\left|\left(U} _{1,\ldots,U} _{s\right)\right.} is given by(\prod ^{r +1} _{i = s +1} \det\left(U ^{a _{i-(p+1)/2}} _{i\right)})/(\beta _{p\left(a} _{s+1} ,\ldots,a _{r+1} \right)\det\left(I _{p-\sum} ^{s} ^{i =1} U ^{\sum ^{r +1} _{i = s +1} a _{i-(p+1)/2}} _{i\right)}) where we write U _{r+1} = I _{p-\sum} ^{rU} ^{i} .

partitioned distribution

Suppose

\left(U _{1,\ldots,U} _{r\right)\sim} D _{p\left(a} _{1,\ldots,a} _{r+1} \right) and suppose that S _{1,\ldots,S} _{t} is a partition of \left[r +1 \right] = \left\{ 1 ,\ldots r +1 \right\} (that is, \cup ^{tS} _{i = \left[r +1 \right]} and S _{i\cap} S _{j=\emptyset} if i ≠ j ). Then, writing U _{(j)} = \sum _{i\in S _{j}} U _{i} and a _{(j)} = \sum _{i\in S _{j}} a _{i} (with U _{r+1} = I _{p-\sum} ^{r} ^{i =1} U _{r} ), we have:\left(U _{(1)} ,\ldots U _{(t)} \right)\sim D _{p\left(a} _{(1)} ,\ldots,a _{(t)} \right).

partitions

Suppose

\left(U _{1,\ldots,U} _{r\right)\sim} D _{p\left(a} _{1,\ldots,a} _{r+1} \right) . DefineU _{i=} \left( \begin{array}{rr} U _{11(i)} & U _{12(i)} \\ U _{21(i)} & U _{22(i)} \end{array} \right) i =1 ,\ldots,r

where

U _{11(i)} is p _{1 x} p _{1} and U _{22(i)} is p _{2 x} p _{2} . Writing the Schur complement U _{22 ⋅} = U _{21(i)} U ^{-1} _{11 (i)} U _{12(i)} we have\left(U _{11(1)} ,\ldots,U _{11(r)} \right)\sim D _{p _{1}} \left(a _{1,\ldots,a} _{r+1} \right) and\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).}

See also

• Inverse Dirichlet distribution

References

A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.