# Flow Models for Arbitrary Conditional Likelihoods

arXiv: https://arxiv.org/abs/1909.06319 · HTML: https://mlsanity.com/paper/1909.06319/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Sep 2019.

## Built on by (most cited first, 30 of 30)

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- [Variational Gibbs Inference for Statistical Model Estimation from Incomplete Data](https://mlsanity.com/paper/2111.13180/) (2021)
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- [Variational Transdimensional Inference](https://mlsanity.com/paper/2506.04749/) (2025)
- [An AI-powered Bayesian Generative Modeling Approach for Arbitrary Conditional Inference](https://mlsanity.com/paper/2601.05355/) (2026)
- [Unsupervised Imputation of Non-ignorably Missing Data Using Importance-Weighted Autoencoders](https://mlsanity.com/paper/2101.07357/) (2021)
- [Semi-supervised Learning with Missing Values Imputation](https://mlsanity.com/paper/2106.01708/) (2021)
- [Viscos Flows: Variational Schur Conditional Sampling With Normalizing Flows](https://mlsanity.com/paper/2107.02474/) (2021)
- [Flow Matching with Missing Data](https://mlsanity.com/paper/2607.28698/) (2026)
- [Evaluator-Dependent Patient-Adaptive ECG Lead-Channel Allocation](https://mlsanity.com/paper/2608.26827/) (2026)
- [RCShift: Certifying When Partial Linkage Suffices for Finite-Sample Decisions](https://mlsanity.com/paper/2609.26207/) (2026)

## Abstract (arXiv)

Understanding the dependencies among features of a dataset is at the core of most unsupervised learning tasks. However, a majority of generative modeling approaches are focused solely on the joint distribution p(x) and utilize models where it is intractable to obtain the conditional distribution of some arbitrary subset of features xu given the rest of the observed covariates xo: p(xu mid xo). Traditional conditional approaches provide a model for a fixed set of covariates conditioned on another fixed set of observed covariates. Instead, in this work we develop a model that is capable of yielding all conditional distributions p(xu mid xo) (for arbitrary xu) via tractable conditional likelihoods. We propose a novel extension of (change of variables based) flow generative models, arbitrary conditioning flow models (AC-Flow), that can be conditioned on arbitrary subsets of observed covariates, which was previously infeasible. We apply AC-Flow to the imputation of features, and also develop a unified platform for both multiple and single imputation by introducing an auxiliary objective that provides a principled single "best guess" for flow models. Extensive empirical evaluations show that our models achieve state-of-the-art performance in both single and multiple imputation across image inpainting and feature imputation in synthetic and real-world datasets. Code is available at https://github.com/lupalab/ACFlow.