text: This paper presents the recurrent estimation of distributions (RED) for modeling real-valued data in a semiparametric fashion. RED models make two novel uses of recurrent neural networks (RNNs) for density estimation of general real-valued data. First, RNNs are used to transform input covariates into a latent space to better capture conditional dependencies in inputs. After, an RNN is used to compute the conditional distributions of the latent covariates. The resulting model is efficient to train, compute, and sample from, whilst producing normalized pdfs. The effectiveness of RED is shown via several real-world data experiments. Our results show that RED models achieve a lower held-out negative log-likelihood than other neural network approaches across multiple dataset sizes and dimensionalities. Further context of the efficacy of RED is provided by considering anomaly detection tasks, where we also observe better performance over alternative models.
  @cite_18: We present Neural Autoregressive Distribution Estimation (NADE) models, which are neural network architectures applied to the problem of unsupervised distribution and density estimation. They leverage the probability product rule and a weight sharing scheme inspired from restricted Boltzmann machines, to yield an estimator that is both tractable and has good generalization performance. We discuss how they achieve competitive performance in modeling both binary and real-valued observations. We also present how deep NADE models can be trained to be agnostic to the ordering of input dimensions used by the autoregressive product rule decomposition. Finally, we also show how to exploit the topological structure of pixels in images using a deep convolutional architecture for NADE.
summary: NADE is a restricted-Boltzmann-machine-inspired density estimator that uses probability product rules and a weight-sharing scheme across conditional densities directly on input covariates @cite_18 . As previously mentioned, a direct application of the chain rule on input dimensions will be sensitive to the order used . To overcome the difficulty of finding a good permutation, deep versions of NADE train the model by sampling random permutations, @math , throughout training. I.e. NADE attempts to find parameters that model the data well (on average) for all permutations. Of course, due to the combinatorial nature of permutations, only a minuscule fraction of permutations will have been explored during training time.