# Arbitrary Conditional Distributions with Energy

arXiv: https://arxiv.org/abs/2102.04426 · HTML: https://mlsanity.com/paper/2102.04426/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 37 references that are in the map.
- First posted on arXiv 8 Feb 2021.

## Builds on (most cited first, 20 of 20)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [GAIN: Missing Data Imputation using Generative Adversarial Nets](https://mlsanity.com/paper/1806.02920/) (2018)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [MisGAN: Learning from Incomplete Data with Generative Adversarial Networks](https://mlsanity.com/paper/1902.09599/) (2019)
- [A Deep and Tractable Density Estimator](https://mlsanity.com/paper/1310.1757/) (2013)
- [Variational Autoencoder with Arbitrary Conditioning](https://mlsanity.com/paper/1806.02382/) (2018)
- [Deep Energy Estimator Networks](https://mlsanity.com/paper/1805.08306/) (2018)
- [Flow Models for Arbitrary Conditional Likelihoods](https://mlsanity.com/paper/1909.06319/) (2019)
- [Transformation Autoregressive Networks](https://mlsanity.com/paper/1801.09819/) (2018)
- [Autoregressive Energy Machines](https://mlsanity.com/paper/1904.05626/) (2019)
- [TraDE: Transformers for Density Estimation](https://mlsanity.com/paper/2004.02441/) (2020)
- [Learning about an exponential amount of conditional distributions](https://mlsanity.com/paper/1902.08401/) (2019)
- [A Universal Marginalizer for Amortized Inference in Generative Models](https://mlsanity.com/paper/1711.00695/) (2017)
- [Estimating conditional density of missing values using deep Gaussian mixture model](https://mlsanity.com/paper/2010.02183/) (2020)

## Built on by (most cited first, 22 of 22)

- [All-in-one simulation-based inference](https://mlsanity.com/paper/2404.09636/) (2024)
- [Training and Inference on Any-Order Autoregressive Models the Right Way](https://mlsanity.com/paper/2205.13554/) (2022)
- [Acquisition Conditioned Oracle for Nongreedy Active Feature Acquisition](https://mlsanity.com/paper/2302.13960/) (2023)
- [A Survey on Active Feature Acquisition Strategies](https://mlsanity.com/paper/2502.11067/) (2025)
- [Any-Order GPT as Masked Diffusion Model: Decoupling Formulation and Architecture](https://mlsanity.com/paper/2506.19935/) (2025)
- [Reviving Any-Subset Autoregressive Models with Principled Parallel Sampling and Speculative Decoding](https://mlsanity.com/paper/2504.20456/) (2025)
- [Generative Marginalization Models](https://mlsanity.com/paper/2310.12920/) (2023)
- [CFMI: Flow Matching for Missing Data Imputation](https://mlsanity.com/paper/2506.09258/) (2025)
- [Posterior Matching for Arbitrary Conditioning](https://mlsanity.com/paper/2201.12414/) (2022)
- [Predictive World Models from Real-World Partial Observations](https://mlsanity.com/paper/2301.04783/) (2023)
- [Deconvolutional Density Network: Modeling Free-Form Conditional Distributions](https://mlsanity.com/paper/2105.14367/) (2021)
- [Distribution Guided Active Feature Acquisition](https://mlsanity.com/paper/2410.03915/) (2024)
- [An AI-powered Bayesian Generative Modeling Approach for Arbitrary Conditional Inference](https://mlsanity.com/paper/2601.05355/) (2026)
- [Learning to sample in Cartesian MRI](https://mlsanity.com/paper/2312.04327/) (2023)
- [Within-basket Recommendation via Neural Pattern Associator](https://mlsanity.com/paper/2401.16433/) (2024)
- [On the connection between Noise-Contrastive Estimation and Contrastive Divergence](https://mlsanity.com/paper/2402.16688/) (2024)
- [Energy-Based Model for Accurate Estimation of Shapley Values in Feature Attribution](https://mlsanity.com/paper/2404.01078/) (2024)
- [Task Expansion and Cross Refinement for Open-World Conditional Modeling](https://mlsanity.com/paper/2603.13308/) (2026)
- [Deep Conditional Measure Quantization](https://mlsanity.com/paper/2301.06907/) (2023)
- [Neural Autoregressive Flows for Markov Boundary Learning](https://mlsanity.com/paper/2603.20791/) (2026)
- [Free Energy Manifold: Score-Based Inference for Hybrid Bayesian Networks](https://mlsanity.com/paper/2605.09839/) (2026)
- [BlockGen: Flexible Blockwise Sequence Modeling with Hybrid Samplers](https://mlsanity.com/paper/2606.02241/) (2026)

## Abstract (arXiv)

Modeling distributions of covariates, or density estimation, is a core challenge in unsupervised learning. However, the majority of work only considers the joint distribution, which has limited utility in practical situations. A more general and useful problem is arbitrary conditional density estimation, which aims to model any possible conditional distribution over a set of covariates, reflecting the more realistic setting of inference based on prior knowledge. We propose a novel method, Arbitrary Conditioning with Energy (ACE), that can simultaneously estimate the distribution p(xu mid xo) for all possible subsets of unobserved features xu and observed features xo. ACE is designed to avoid unnecessary bias and complexity -- we specify densities with a highly expressive energy function and reduce the problem to only learning one-dimensional conditionals (from which more complex distributions can be recovered during inference). This results in an approach that is both simpler and higher-performing than prior methods. We show that ACE achieves state-of-the-art for arbitrary conditional likelihood estimation and data imputation on standard benchmarks.