Yoel Zeldes thinks about neural networks from a different perspective:

The term

logP(w), which represents our prior, acts as a regularization term. Choosing a Gaussian distribution with mean 0 as the prior, you’ll get the mathematical equivalence of L2 regularization.Now that we start thinking about neural networks as probabilistic creatures, we can let the fun begin. For start, who says we have to output one set of weights at the end of the training process? What if instead of learning the model’s weights, we learn a distribution over the weights? This will allow us to estimate uncertainty over the weights. So how do we do that?

It’s an interesting approach to the problem.