Homework 9

Let $X = [X_1,X_2,X_3]^T$ where $X_1,X_2,X_3$ are all binary with outcomes 0 or 1. We consider two models for $X$.  In model 1, we consider all distributions for $X$ whose probability mass function factors as 

$p(x_1,x_2,x_3) = p(x_1)p(x_2)p(x_3 \mid x_1,x_2)$ for all outcomes of $X$. 

In model 2, we consider all distributions for $X$ whose probability mass function factors as 

$p(x_1,x_2,x_3) = p(x_1)p(x_2 \mid x_3)p(x_3)$for all outcomes of $X$. 

a) Give the matrices $A_1,A_2$ that parametrize models 1 and 2 (respectively) as log-linear models $\mathcal{M}_{A_1}, \mathcal{M}_{A_2}$.

b) Is it true that $\mathcal{M}_{A_1} = \mathcal{M}_{A_2}$?  Can you give a proof of this using your prior knowledge from statistics courses?  Can you prove it using algebraic geometry methods?