  Neural autoregressive models are explicit density estimators that achieve
state-of-the-art likelihoods for generative modeling. The D-dimensional data
distribution is factorized into an autoregressive product of one-dimensional
conditional distributions according to the chain rule. Data completion is a
more involved task than data generation: the model must infer missing variables
for any partially observed input vector. Previous work introduced an
order-agnostic training procedure for data completion with autoregressive
models. Missing variables in any partially observed input vector can be imputed
efficiently by choosing an ordering where observed dimensions precede
unobserved ones and by computing the autoregressive product in this order. In
this paper, we provide evidence that the order-agnostic (OA) training procedure
is suboptimal for data completion. We propose an alternative procedure (OA++)
that reaches better performance in fewer computations. It can handle all data
completion queries while training fewer one-dimensional conditional
distributions than the OA procedure. In addition, these one-dimensional
conditional distributions are trained proportionally to their expected usage at
inference time, reducing overfitting. Finally, our OA++ procedure can exploit
prior knowledge about the distribution of inference completion queries, as
opposed to OA. We support these claims with quantitative experiments on
standard datasets used to evaluate autoregressive generative models.
