Aims. We present an innovative artificial neural network (ANN) architecture, called Generative ANN (GANN), that computes the forward model, that is it learns the function that relates the unknown outputs (stellar atmospheric parameters, in this case) to the given inputs (spectra). Such a model can be integrated in a Bayesian framework to estimate the posterior distribution of the outputs. Methods. The architecture of the GANN follows the same scheme as a normal ANN, but with the inputs and outputs inverted. We train the network with the set of atmospheric parameters (Teff, logg, [Fe/H] and [alpha/Fe]), obtaining the stellar spectra for such inputs. The residuals between the spectra in the grid and the estimated spectra are minimized using a validation dataset to keep solutions as general as possible. Results. The performance of both conventional ANNs and GANNs to estimate the stellar parameters as a function of the star brightness is presented and compared for different Galactic populations. GANNs provide significantly improved parameterizations for early and intermediate spectral types with rich and intermediate metallicities. The behaviour of both algorithms is very similar for our sample of late-type stars, obtaining residuals in the derivation of [Fe/H] and [alpha/Fe] below 0.1dex for stars with Gaia magnitude Grvs<12, which accounts for a number in the order of four million stars to be observed by the Radial Velocity Spectrograph of the Gaia satellite. Conclusions. Uncertainty estimation of computed astrophysical parameters is crucial for the validation of the parameterization itself and for the subsequent exploitation by the astronomical community. GANNs produce not only the parameters for a given spectrum, but a goodness-of-fit between the observed spectrum and the predicted one for a given set of parameters. Moreover, they allow us to obtain the full posterior distribution...