  Simulation-based inference (SBI) provides a powerful framework for inferring
posterior distributions of stochastic simulators in a wide range of domains. In
many settings, however, the posterior distribution is not the end goal itself
-- rather, the derived parameter values and their uncertainties are used as a
basis for deciding what actions to take. Unfortunately, because posterior
distributions provided by SBI are (potentially crude) approximations of the
true posterior, the resulting decisions can be suboptimal. Here, we address the
question of how to perform Bayesian decision making on stochastic simulators,
and how one can circumvent the need to compute an explicit approximation to the
posterior. Our method trains a neural network on simulated data and can predict
the expected cost given any data and action, and can, thus, be directly used to
infer the action with lowest cost. We apply our method to several benchmark
problems and demonstrate that it induces similar cost as the true posterior
distribution. We then apply the method to infer optimal actions in a real-world
simulator in the medical neurosciences, the Bayesian Virtual Epileptic Patient,
and demonstrate that it allows to infer actions associated with low cost after
few simulations.
