Wake-sleep algorithm

During wake, present an observed input and sample its hidden representation layer by layer through the bottom-up recognition distribution q_phi(h|x). Hold recognition parameters fixed for this update and ascend log p_theta(x,h) in the generative parameters. For the original Bernoulli-sigmoid units, a generative weight receives a learning-rate-scaled product of its sampled parent state and the difference between the sampled target state and the generative activation probability; biases use a constant parent. During sleep, sample a full configuration (x,h) ancestrally from the current top-down generative model, then ascend log q_phi(h|x) in the recognition parameters while holding generative parameters fixed. The generated lower-layer states serve as inputs and their known sampled upper-layer causes as targets. This also gives local Bernoulli delta rules in the original parameterization. The wake update follows the generative-parameter variational-bound gradient at fixed q; the sleep update instead fits recognition on model-generated inputs and targets the posterior-to-recognition KL direction averaged under model inputs. The two phases are not generally gradients of the same data ELBO.