  Fitting generative models to sequential data typically involves two recursive
computations through time, one forward and one backward. The latter could be a
computation of the loss gradient (as in backpropagation through time), or an
inference algorithm (as in the RTS/Kalman smoother). The backward pass in
particular is computationally expensive (since it is inherently serial and
cannot exploit GPUs), and difficult to map onto biological processes.
Work-arounds have been proposed; here we explore a very different one:
requiring the generative model to learn the joint distribution over current and
previous states, rather than merely the transition probabilities. We show on
toy datasets that different architectures employing this principle can learn
aspects of the data typically requiring the backward pass.
