  Neural nets have become popular to accelerate parameter inferences,
especially for the upcoming generation of galaxy surveys in cosmology. As
neural nets are approximative by nature, a recurrent question has been how to
propagate the neural net's approximation error, in order to avoid biases in the
parameter inference. We present a Bayesian solution to propagating a neural
net's approximation error and thereby debiasing parameter inference. We exploit
that a neural net reports its approximation errors during the validation phase.
We capture the thus reported approximation errors via the highest-order summary
statistics, allowing us to eliminate the neural net's bias during inference,
and propagating its uncertainties. We demonstrate that our method is quickly
implemented and successfully infers parameters even for strongly biased neural
nets. In summary, our method provides the missing element to judge the accuracy
of a posterior if it cannot be computed based on an infinitely accurately
theory code.
