In statistics, the matrix variate Dirichlet distribution is a generalization of the matrix variate beta distribution and of the Dirichlet distribution.
Suppose
are
positive definite matrices with
also positive-definite, where
is the
identity matrix. Then we say that the
have a matrix variate Dirichlet distribution,
, if their joint probability density function is
![{\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}^{r}U_{i}\right)^{a_{r+1}-(p+1)/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e46e75b0604681a8e86ba6c537f8d86460810fe)
where
and
is the multivariate beta function.
If we write
then the PDF takes the simpler form
![{\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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c35063d5bb6dda2e1aeb097ae211feb0442fad8)
on the understanding that
.
Theorems
generalization of chi square-Dirichlet result
Suppose
are independently distributed Wishart
positive definite matrices. Then, defining
(where
is the sum of the matrices and
is any reasonable factorization of
), we have
![{\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(n_{1}/2,...,n_{r+1}/2\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82ce2c19c55843c482678f24a535df2fdda88c53)
Marginal distribution
If
, and if
, then:
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a54154d885753f47b35ed8f9b372a5c0322ac7f9)
Conditional distribution
Also, with the same notation as above, the density of
is given by
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/344da901ae33d0f9ebd479b4c1f554196d708c40)
where we write
.
partitioned distribution
Suppose
and suppose that
is a partition of
(that is,
and
if
). Then, writing
and
(with
), we have:
![{\displaystyle \left(U_{(1)},\ldots U_{(t)}\right)\sim D_{p}\left(a_{(1)},\ldots ,a_{(t)}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5281a346a3fd7718207dd8b4a3390ef98f9ab6)
partitions
Suppose
. Define
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/210f8e9bdbc95eb591ac1e31437461406fe3a9df)
where
is
and
is
. Writing the Schur complement
we have
![{\displaystyle \left(U_{11(1)},\ldots ,U_{11(r)}\right)\sim D_{p_{1}}\left(a_{1},\ldots ,a_{r+1}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e945294eab29daad2451cafa40aba9edebde7ac7)
and
![{\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_{1}r/2\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79befd7cf0722c9a6ff85193ab2e4e3f643a9923)
See also
- Inverse Dirichlet distribution
References
A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.