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A Deep Belief Network is a multi-layered, feed-forward network in which each successive layer infers about latent variables of the input from the output of its preceding layers.

ANN implementing DBN have been around for a long time (they go back at least to Fukushima's Neocognitron).

Hinton proposes building deep belief networks by stacking RBMs and training them unsupervised and in ascending order. After that, the network goes into feed-forward mode and backprop can be used to learn the actual task. Thus, some of the problems of backprop are solved by initializing the weights via unsupervised learning.