Generative Model: Autoregressive Model

An autoregressive (AR) model is autoregressive,

$$ \begin{equation} \log p_\theta (x) = \sum_{t=1}^T \log p_\theta ( x_{t} \mid {x_{<t}} ). \end{equation} $$

In the above example, the likelihood is modeled as

$$ \begin{align} p_\theta (x) &= \Pi_{t=1}^T p_\theta (x_t \mid x_{1:t-1}) \\ &= p_\theta(x_2 \mid x_{1:1}) p_\theta(x_3 \mid x_{1:2}) \cdots p_\theta(x_T \mid x_{1:T-1}) \end{align} $$

Taking the log of it

$$ \ln p_\theta (x) = \sum_{t=1}^T \ln p_\theta (x_t \mid x_{1:t-1}) $$

Notations and Conventions

In AR models, we have to mention the preceding nodes (${x_{<t}}$) of a specific node ($x_{t}$). For $t=5$, the relations between ${x_{<5}}$ and $x_5$ is shown in the following illustration.

There are different notations for such relations.

  • In Uria et al., the authors use $p(x_{o_d}\mid \mathbf x_{o_{<d}})$ [^Uria2016].
  • In Liu et al. and Papamakarios et al., the authors use $p(x_{t}\mid \mathbf x_{1:t-1})$ [^Liu2020][^Papamakarios2017].
  • In Germain et al., the authors use $p(x_t\mid \mathbf x_{<t})$ [^Germain2015].

In the current review, we expanded the vector notation $\mathbf x_{<t}$ into a set notation as it is not necessarily a vector.

Planted: by ;

  1. Uria2016 Uria B, Côté M-A, Gregor K, Murray I, Larochelle H. Neural Autoregressive Distribution Estimation. arXiv [cs.LG]. 2016. Available:
  2. Triebe2019 Triebe O, Laptev N, Rajagopal R. AR-Net: A simple Auto-Regressive Neural Network for time-series. arXiv [cs.LG]. 2019. Available:
  3. Ho2019 Ho G. George Ho. In: Eigenfoo [Internet]. 9 Mar 2019 [cited 19 Sep 2021]. Available:
  4. Papamakarios2017 Papamakarios G, Pavlakou T, Murray I. Masked Autoregressive Flow for Density Estimation. arXiv [stat.ML]. 2017. Available:
  5. Germain2015 Germain M, Gregor K, Murray I, Larochelle H. MADE: Masked autoencoder for distribution estimation. 32nd International Conference on Machine Learning, ICML 2015. 2015;2: 881–889. Available:
  6. Liu2020 Liu X, Zhang F, Hou Z, Wang Z, Mian L, Zhang J, et al. Self-supervised Learning: Generative or Contrastive. arXiv [cs.LG]. 2020. Available:
  7. Lippe Lippe P. Tutorial 12: Autoregressive Image Modeling — UvA DL Notebooks v1.1 documentation. In: UvA Deep Learning Tutorials [Internet]. [cited 20 Sep 2021]. Available:
  8. rogen rogen-george. rogen-george/Deep-Autoregressive-Model. In: GitHub [Internet]. [cited 20 Sep 2021]. Available:
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L Ma (2021). 'Generative Model: Autoregressive Model', Datumorphism, 08 April. Available at: