Figure 13.

The Free Energy as certainty-equivalent (Example 5). (a) Utility function U as a function of $z_i$ (top) and expected utilities for the coarse-grained partitions $\{z_1,z_2\}$ and $\{z_3,z_4\}$ corresponding to the choices $x_1$ and $x_2$, respectively, for a bounded-rational decision-maker with $\beta =0.9$ (bottom). (b) The bounded optimal probability distribution over $x_i$ (top) does not correspond to the expected utilities in (a) but to the Free Energy of the second decision-step, i.e., the decision about Y given X (bottom).