Ex. 17.10

Ex. 17.10

Using a simple binary graphical model with just two variables, show why it is essential to include a constant node $X_0\equiv 1$ in the model.

Soln. 17.10

Suppose we only have two variables $X_1$ and $X_2$. When $X_1$ and $X_2$ are not connected, the joint probability in (17.28) reduces to a constant 1. When a constant node $X_0\equiv 1$ is included and is always connected to $X_1$ and $X_2$, then the join probability becomes

$$\begin{array}{rcl}p(\mathbf{\text{X}},\mathbf{\Theta })& =& \exp ({\theta }_{01}{X}_1+{\theta }_{02}{X}_2-\log (\exp ({\theta }_{01})+\exp ({\theta }_{02}+\exp ({\theta }_{01}+{\theta }_{02}))))\\ & =& \exp ({\theta }_{01}{X}_1+{\theta }_{02}{X}_2)/[1+\exp ({\theta }_{01})+\exp ({\theta }_{02})+\exp ({\theta }_{01}+{\theta }_{02})].\end{array}$$