# Conditional Density Estimation with Bayesian Normalising Flows

arXiv: https://arxiv.org/abs/1802.04908 · HTML: https://mlsanity.com/paper/1802.04908/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2018.

## Built on by (most cited first, 30 of 54)

- [Low-Light Image Enhancement with Normalizing Flow](https://mlsanity.com/paper/2109.05923/) (2021)
- [VCNet and Functional Targeted Regularization For Learning Causal Effects of Continuous Treatments](https://mlsanity.com/paper/2103.07861/) (2021)
- [High Fidelity Image Counterfactuals with Probabilistic Causal Models](https://mlsanity.com/paper/2306.15764/) (2023)
- [Probabilistic Conformal Prediction Using Conditional Random Samples](https://mlsanity.com/paper/2206.06584/) (2022)
- [Counterfactual Identifiability of Bijective Causal Models](https://mlsanity.com/paper/2302.02228/) (2023)
- [Rhino: Deep Causal Temporal Relationship Learning With History-dependent Noise](https://mlsanity.com/paper/2210.14706/) (2022)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Benchmarking Counterfactual Image Generation](https://mlsanity.com/paper/2403.20287/) (2024)
- [Normalizing Flows for Interventional Density Estimation](https://mlsanity.com/paper/2209.06203/) (2022)
- [Improvement-Focused Causal Recourse (ICR)](https://mlsanity.com/paper/2210.15709/) (2022)
- [Bounds on Representation-Induced Confounding Bias for Treatment Effect Estimation](https://mlsanity.com/paper/2311.11321/) (2023)
- [Conformal Prediction for Causal Effects of Continuous Treatments](https://mlsanity.com/paper/2407.03094/) (2024)
- [Reliable Off-Policy Learning for Dosage Combinations](https://mlsanity.com/paper/2305.19742/) (2023)
- [Mitigating attribute amplification in counterfactual image generation](https://mlsanity.com/paper/2403.09422/) (2024)
- [Quantifying Aleatoric Uncertainty of the Treatment Effect: A Novel Orthogonal Learner](https://mlsanity.com/paper/2411.03387/) (2024)
- [A generative flow for conditional sampling via optimal transport](https://mlsanity.com/paper/2307.04102/) (2023)
- [A Kernel-Based Conditional Two-Sample Test Using Nearest Neighbors (with Applications to Calibration, Regression Curves, and Simulation-Based Inference)](https://mlsanity.com/paper/2407.16550/) (2024)
- [Conditional Measurement Density Estimation in Sequential Monte Carlo via Normalizing Flow](https://mlsanity.com/paper/2203.08617/) (2022)
- [Short-Term Density Forecasting of Low-Voltage Load using Bernstein-Polynomial Normalizing Flows](https://mlsanity.com/paper/2204.13939/) (2022)
- [Efficient SAGE Estimation via Causal Structure Learning](https://mlsanity.com/paper/2304.03113/) (2023)
- [Conditional Density Estimation with Histogram Trees](https://mlsanity.com/paper/2410.11449/) (2024)
- [Generative models for decision-making under distributional shift](https://mlsanity.com/paper/2604.04342/) (2026)
- [Flow-based Spatio-Temporal Structured Prediction of Motion Dynamics](https://mlsanity.com/paper/2104.04391/) (2021)
- [Deconvolutional Density Network: Modeling Free-Form Conditional Distributions](https://mlsanity.com/paper/2105.14367/) (2021)
- [TutteNet: Injective 3D Deformations by Composition of 2D Mesh Deformations](https://mlsanity.com/paper/2406.12121/) (2024)
- [Harnessing Synthetic Data from Generative AI for Statistical Inference](https://mlsanity.com/paper/2603.05396/) (2026)
- [Uncertainty estimation under model misspecification in neural network regression](https://mlsanity.com/paper/2111.11763/) (2021)
- [Graph network for learning bi-directional physics](https://mlsanity.com/paper/2112.07054/) (2021)
- [Conditional Kernel Imitation Learning for Continuous State Environments](https://mlsanity.com/paper/2308.12573/) (2023)
- [Generative Modeling: A Review](https://mlsanity.com/paper/2501.05458/) (2024)

## Abstract (arXiv)

Modeling complex conditional distributions is critical in a variety of settings. Despite a long tradition of research into conditional density estimation, current methods employ either simple parametric forms or are difficult to learn in practice. This paper employs normalising flows as a flexible likelihood model and presents an efficient method for fitting them to complex densities. These estimators must trade-off between modeling distributional complexity, functional complexity and heteroscedasticity without overfitting. We recognize these trade-offs as modeling decisions and develop a Bayesian framework for placing priors over these conditional density estimators using variational Bayesian neural networks. We evaluate this method on several small benchmark regression datasets, on some of which it obtains state of the art performance. Finally, we apply the method to two spatial density modeling tasks with over 1 million datapoints using the New York City yellow taxi dataset and the Chicago crime dataset.