We present a framework for training artificial neural networks (ANNs) as surrogate Bayesian models for the inference of plasma parameters from diagnostic data collected at nuclear fusion experiments, with the purpose of providing a fast approximation of conventional Bayesian inference. Because of the complexity of the models involved, conventional Bayesian inference can require tens of minutes for analyzing one single measurement, while hundreds of thousands can be collected during a single plasma discharge. The ANN surrogates can reduce the analysis time down to tens/hundreds of microseconds per single measurement. The core idea is to generate the training data by sampling them from the joint probability distribution of the parameters and observations of the original Bayesian model. The network can be trained to learn the reconstruction of plasma parameters from observations and the model joint probability distribution from plasma parameters and observations. Previous work has validated the application of such a framework to the former case at the Wendelstein 7-X and Joint European Torus experiments. Here, we first give a description of the general methodological principles allowing us to generate the training data, and then we show an example application of the reconstruction of the joint probability distribution of an effective ion charge Zeff-bremsstrahlung model from data collected at the latest W7-X experimental campaign. One key feature of such an approach is that the network is trained exclusively on data generated with the Bayesian model, requiring no experimental data. This allows us to replicate the training scheme and generate fast, surrogate ANNs for any validated Bayesian diagnostic model.