#ising-model

The Ising model $p(x_1, x_2; \theta) \propto \exp(\theta x_1 x_2 + x_1 + x_2)$ with $x_i \in \{-1, 1\}$ has $Z(\theta) = 2e^{-\theta} + e^{\theta + 2} + e^{\theta - 2}$. The figure shows $f(\theta) = \partial \log Z(\theta)/\partial\theta$. You observe the three data points $(x_1, x_2) = (-1, -1)$, $(-1, 1)$ and $(1, -1)$. Read the maximum likelihood estimate $\hat\theta$ off the figure.