MAP Criterion

The expected field $\hat{x}$ is such that :

$$\hat{x} = \arg \max_x (P(c\vert x)P(i\vert x)P(x))$$

Assuming :

$P(c\vert x)=\frac{\exp (-U_c(x))}{Z_c}$, $P(i\vert x)=\frac{\exp (-U_i(x))}{Z_i}$ and $P(x)=\frac{\exp(-U_{ap}(x))}{Z_{ap}}$,

Then :

$$\hat{x} = \arg \min_x (U_c(x)+U_i(x)+U_{ap}(x))$$