# Stepwise Variational Inference with Vine Copulas

arXiv: https://arxiv.org/abs/2603.22959 · HTML: https://mlsanity.com/paper/2603.22959/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 65 references that are in the map.
- First posted on arXiv 24 Mar 2026.

## Builds on (most cited first, 28 of 28)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [Rank-normalization, folding, and localization: An improved widehatR for assessing convergence of MCMC](https://mlsanity.com/paper/1903.08008/) (2019)
- [Variational Boosting: Iteratively Refining Posterior Approximations](https://mlsanity.com/paper/1611.06585/) (2016)
- [Automatic Variational Inference in Stan](https://mlsanity.com/paper/1506.03431/) (2015)
- [Boosting Variational Inference](https://mlsanity.com/paper/1611.05559/) (2016)
- [Robust, Accurate Stochastic Optimization for Variational Inference](https://mlsanity.com/paper/2009.00666/) (2020)
- [Copulas as High-Dimensional Generative Models: Vine Copula Autoencoders](https://mlsanity.com/paper/1906.05423/) (2019)
- [Alpha-divergence Variational Inference Meets Importance Weighted Auto-Encoders: Methodology and Asymptotics](https://mlsanity.com/paper/2210.06226/) (2022)
- [Boosting Black Box Variational Inference](https://mlsanity.com/paper/1806.02185/) (2018)
- [Universal Boosting Variational Inference](https://mlsanity.com/paper/1906.01235/) (2019)
- [Variational Gaussian Copula Inference](https://mlsanity.com/paper/1506.05860/) (2015)
- [Variational Inference for Uncertainty Quantification: an Analysis of Trade-offs](https://mlsanity.com/paper/2403.13748/) (2024)
- [Boosting Variational Inference: an Optimization Perspective](https://mlsanity.com/paper/1708.01733/) (2017)
- [Retrospective Uncertainties for Deep Models using Vine Copulas](https://mlsanity.com/paper/2302.12606/) (2023)
- [Quasi-Bayes meets Vines](https://mlsanity.com/paper/2406.12764/) (2024)
- [Flexible Variational Bayes based on a Copula of a Mixture](https://mlsanity.com/paper/2106.14392/) (2021)
- [Implicit copula variational inference](https://mlsanity.com/paper/2111.09511/) (2021)
- [Towards a universal representation of statistical dependence](https://mlsanity.com/paper/2302.08151/) (2023)
- [Wild posteriors in the wild](https://mlsanity.com/paper/2503.00239/) (2025)
- [Vine Copulas as Differentiable Computational Graphs](https://mlsanity.com/paper/2506.13318/) (2025)
- [Vector Copula Variational Inference and Dependent Block Posterior Approximations](https://mlsanity.com/paper/2503.01072/) (2025)

## Abstract (arXiv)

We propose stepwise variational inference (VI) with vine copulas: a universal VI procedure that combines vine copulas with a novel stepwise estimation procedure of the variational parameters. Vine copulas consist of a nested sequence of trees built from copulas, where more complex latent dependence can be modeled with increasing number of trees. We propose to estimate the vine copula approximate posterior in a stepwise fashion, tree by tree along the vine structure. Further, we show that the usual backward Kullback-Leibler divergence cannot recover the correct parameters in the vine copula model, thus the evidence lower bound is defined based on the R'enyi divergence. Finally, an intuitive stopping criterion for adding further trees to the vine eliminates the need to pre-define a complexity parameter of the variational distribution, as required for most other approaches. Thus, our method interpolates between mean-field VI (MFVI) and full latent dependence. In many applications, in particular sparse Gaussian processes, our method is parsimonious with parameters, while outperforming MFVI.