pymc.WishartBartlett#
- pymc.WishartBartlett(name, S, nu, is_cholesky=False, return_cholesky=False, initval=None)[source]#
Bartlett decomposition of the Wishart distribution.
As the Wishart distribution requires the matrix to be symmetric positive semi-definite, it is impossible for MCMC to ever propose acceptable matrices.
Instead, we can use the Barlett decomposition which samples a lower diagonal matrix. Specifically:
\[ \begin{align}\begin{aligned}\begin{split}\text{If} L \sim \begin{pmatrix} \sqrt{c_1} & 0 & 0 \\ z_{21} & \sqrt{c_2} & 0 \\ z_{31} & z_{32} & \sqrt{c_3} \end{pmatrix}\end{split}\\\begin{split}\text{with} c_i \sim \chi^2(n-i+1) \text{ and } n_{ij} \sim \mathcal{N}(0, 1), \text{then} \\ L \times A \times A.T \times L.T \sim \text{Wishart}(L \times L.T, \nu)\end{split}\end{aligned}\end{align} \]See http://en.wikipedia.org/wiki/Wishart_distribution#Bartlett_decomposition for more information.
- Parameters:
- S
ndarray p x p positive definite matrix Or: p x p lower-triangular matrix that is the Cholesky factor of the covariance matrix.
- nutensor_like of
int Degrees of freedom, > dim(S).
- is_choleskybool, default=False
Input matrix S is already Cholesky decomposed as S.T * S
- return_choleskybool, default=False
Only return the Cholesky decomposed matrix.
- initval
ndarray p x p positive definite matrix used to initialize
- S
Notes
This is not a standard Distribution class but follows a similar interface. Besides the Wishart distribution, it will add RVs name_c and name_z to your model which make up the matrix.
This distribution is usually a bad idea to use as a prior for multivariate normal. You should instead use LKJCholeskyCov or LKJCorr.