I've been reading some papers on Bayesian Neural Networks and one that caught my attention is titled Deep Neural Networks as Gaussian Processes where they use Gaussian priors for the neural network weights and end up with a Gaussian Process, given that the hidden layer is wide enough.
By using the GP prior over functions produced by the network they are then able to perform Bayesian inference for regression tasks using deep neural networks.
My question is: What is the benefit gained from this? Why not simply use a vanilla GP? Also, is there some advantage of their method over using popular methods such as MCMC or variational inference to approximate the posterior distribution of the network weights?
My knowledge on Bayesian Neural Networks is still green, so any clarity on this topic would be greatly appreciated.