Ex. 17.11

Ex. 17.11

Show that the Ising model (17.28) (17.28) for the joint probabilities in a discrete graphical model implies that the conditional distributions have the logistic form (17.30).

Soln. 17.11

The claim follows from Bayesian formula. Note that we include constant node \(X_0\equiv 1\) which connects to each \(X_i\). By (17.28) in the text, We have

\[\begin{eqnarray} &&\text{Pr}(X_j=1|X_{-j}=x_{-j})\non\\ &=& \frac{\text{Pr}(X_j=1, X_{-j}=x_{-j})}{\text{Pr}(X_j=1, X_{-j}=x_{-j}) + \text{Pr}(X_j=0, X_{-j}=x_{-j})}\non\\ &=&\frac{\exp\left(\theta_{j0} + \sum_{(j,k)\in E}\theta_{jk}x_k + \sum_{\substack{(i,k)\in E\\ i, k \neq j}}\theta_{ik}x_ix_k\right)}{\exp\left(\theta_{j0} + \sum_{(j,k)\in E}\theta_{jk}x_k + \sum_{\substack{(i,k)\in E\\ i, k \neq j}}\theta_{ik}x_ix_k\right) + \exp\left(\sum_{\substack{(i,k)\in E\\ i, k \neq j}}\theta_{ik}x_ix_k\right)}\non\\ &=&\frac{1}{1+\exp\left(-\theta_{j0}-\sum_{(j,k)\in E}\theta_{jk}x_k\right)}.\non \end{eqnarray}\]