  Neural Autoregressive Distribution Estimators (NADEs) have recently been
shown as successful alternatives for modeling high dimensional multimodal
distributions. One issue associated with NADEs is that they rely on a
particular order of factorization for $P(\mathbf{x})$. This issue has been
recently addressed by a variant of NADE called Orderless NADEs and its deeper
version, Deep Orderless NADE. Orderless NADEs are trained based on a criterion
that stochastically maximizes $P(\mathbf{x})$ with all possible orders of
factorizations. Unfortunately, ancestral sampling from deep NADE is very
expensive, corresponding to running through a neural net separately predicting
each of the visible variables given some others. This work makes a connection
between this criterion and the training criterion for Generative Stochastic
Networks (GSNs). It shows that training NADEs in this way also trains a GSN,
which defines a Markov chain associated with the NADE model. Based on this
connection, we show an alternative way to sample from a trained Orderless NADE
that allows to trade-off computing time and quality of the samples: a 3 to
10-fold speedup (taking into account the waste due to correlations between
consecutive samples of the chain) can be obtained without noticeably reducing
the quality of the samples. This is achieved using a novel sampling procedure
for GSNs called annealed GSN sampling, similar to tempering methods that
combines fast mixing (obtained thanks to steps at high noise levels) with
accurate samples (obtained thanks to steps at low noise levels).
