Homework 7

a) Consider the vector $h = (1,1,1,2,2,2)$ and the matrix 

                            $A = \begin{pmatrix} 2 & 0 & 0 & 1 & 1 & 0 \\ 0 & 2 & 0 & 1 & 0 & 1\\ 0 & 0 & 2 & 0 & 1 & 1 \end{pmatrix}$.

         (i) Compute the generators for the toric ideals $I_A$ and $I_{A,h}$.

         (ii) What familiar statistical model is the discrete exponential family $\mathcal{M}_{A,h}$?  (Be sure to justify your answer.)

b) Consider the linear space $L$ of the $3 \times 3$ real symmetric matrices $\mathbb{S}^{3 \times 3}$ defined by 

                            $L = \{ K \in \mathbb{S}^{3\times 3} : k_{22} = k_{33}, k_{13} = 0\}$.

         (i) Determine the generators of the vanishing ideal $I(M)$ in $\mathbb{R}[\Sigma] = \mathbb{R}[\sigma_{11},\sigma_{22},\sigma_{33},\sigma_{12},\sigma_{13},\sigma_{23}]$ where 

                            $M = \{ \Sigma\in \mathbb{S}^{3\times 3} : \Sigma^{-1}\in L\}$.

         (ii) Give a statistical interpretation of the generators of $I(M)$ in terms of the Gaussian exponential family

                            $\mathcal{M} = \{ \mathbf{X} = [X_1,X_2,X_3]^T\sim \textrm{N}(\mathbf{0}, \Sigma) : \Sigma \in M \cap \textrm{PD}_3\}$,

              where $\textrm{PD}_3$ denotes the cone of positive definite matrices in $\mathbb{S}^{3 \times 3}$.