Algorithm 1 GBC for MEU

Simulate ${(Y^{\left(i\right)},{\theta }^{\left(i\right)})}_{1\le i\le N}\sim p(y\mid \theta )$ or $Y^{\left(i\right)}=f\left({\theta }^{\left(i\right)}\right)$ and ${\theta }^{\left(i\right)}\sim \pi \left(\theta \right)$.   Simulate the utility $u^{\left(i\right)}=U(d^{\left(i\right)},Y^{\left(i\right)},{\theta }^{\left(i\right)})$   Train H using the simulated dataset for $i=1,\dots N$, via ${\widehat{\theta }}^{\left(i\right)}=H(Y^{\left(i\right)},z^{\left(i\right)})$   Train U using the simulated dataset $U_d=U(H{(S\left(Y^{\left(i\right)}\right),z)}^{\left(i\right)},d)$ for $i=1,\dots N$   Pick a decision d that maximizes the expected utility. We use Monte Carlo to estimate the expected utility. $$E\left(U_d\right)=\sum _{i=1}^N{F}_{U_d}^{-1}\left(u_i\right)\to \underset{d}{\mathrm{maximize}}$$