# Variational Inference with Continuously-Indexed Normalizing Flows

arXiv: https://arxiv.org/abs/2007.05426 · HTML: https://mlsanity.com/paper/2007.05426/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jul 2020.

## Built on by (most cited first, 15 of 15)

- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Towards Understanding the Dynamics of Gaussian-Stein Variational Gradient Descent](https://mlsanity.com/paper/2305.14076/) (2023)
- [On Incorporating Inductive Biases into VAEs](https://mlsanity.com/paper/2106.13746/) (2021)
- [Continuous Latent Process Flows](https://mlsanity.com/paper/2106.15580/) (2021)
- [Automatic variational inference with cascading flows](https://mlsanity.com/paper/2102.04801/) (2021)
- [Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction](https://mlsanity.com/paper/2106.04170/) (2021)
- [Deterministic training of generative autoencoders using invertible layers](https://mlsanity.com/paper/2205.09546/) (2022)
- [Promoting Ensemble Diversity with Interactive Bayesian Distributional Robustness for Fine-tuning Foundation Models](https://mlsanity.com/paper/2506.07247/) (2025)
- [Self-reinforced polynomial approximation methods for concentrated probability densities](https://mlsanity.com/paper/2303.02554/) (2023)
- [Ergodic Generative Flows](https://mlsanity.com/paper/2505.03561/) (2025)
- [Diffusion models for probabilistic programming](https://mlsanity.com/paper/2311.00474/) (2023)
- [A Method of Moments Embedding Constraint and its Application to Semi-Supervised Learning](https://mlsanity.com/paper/2404.17978/) (2024)
- [Discretely Indexed Flows](https://mlsanity.com/paper/2204.01361/) (2022)
- [Binary classification based Monte Carlo simulation](https://mlsanity.com/paper/2307.16035/) (2023)
- [Learning a Mixture of GFlowNets](https://mlsanity.com/paper/2610.07562/) (2026)

## Abstract (arXiv)

Continuously-indexed flows (CIFs) have recently achieved improvements over baseline normalizing flows on a variety of density estimation tasks. CIFs do not possess a closed-form marginal density, and so, unlike standard flows, cannot be plugged in directly to a variational inference (VI) scheme in order to produce a more expressive family of approximate posteriors. However, we show here how CIFs can be used as part of an auxiliary VI scheme to formulate and train expressive posterior approximations in a natural way. We exploit the conditional independence structure of multi-layer CIFs to build the required auxiliary inference models, which we show empirically yield low-variance estimators of the model evidence. We then demonstrate the advantages of CIFs over baseline flows in VI problems when the posterior distribution of interest possesses a complicated topology, obtaining improved results in both the Bayesian inference and surrogate maximum likelihood settings.