# The Shrinkage-Delinkage Trade-off: An Analysis of Factorized Gaussian Approximations for Variational Inference

arXiv: https://arxiv.org/abs/2302.09163 · HTML: https://mlsanity.com/paper/2302.09163/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 26 references that are in the map.
- First posted on arXiv 17 Feb 2023.

## Builds on (most cited first, 6 of 6)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [Covariances, Robustness, and Variational Bayes](https://mlsanity.com/paper/1709.02536/) (2017)
- [Pathfinder: Parallel quasi-Newton variational inference](https://mlsanity.com/paper/2108.03782/) (2021)
- [Amortized Variational Inference for Simple Hierarchical Models](https://mlsanity.com/paper/2111.03144/) (2021)
- [The Kernel Interaction Trick: Fast Bayesian Discovery of Pairwise Interactions in High Dimensions](https://mlsanity.com/paper/1905.06501/) (2019)

## Built on by (most cited first, 10 of 10)

- [Amortized Variational Inference: When and Why?](https://mlsanity.com/paper/2307.11018/) (2023)
- [Black Box Variational Inference with a Deterministic Objective: Faster, More Accurate, and Even More Black Box](https://mlsanity.com/paper/2304.05527/) (2023)
- [Amortized Bayesian Multilevel Models](https://mlsanity.com/paper/2408.13230/) (2024)
- [Variational Inference for Uncertainty Quantification: an Analysis of Trade-offs](https://mlsanity.com/paper/2403.13748/) (2024)
- [Variational Inference in Location-Scale Families: Exact Recovery of the Mean and Correlation Matrix](https://mlsanity.com/paper/2410.11067/) (2024)
- [Generalized Guarantees for Variational Inference in the Presence of Even and Elliptical Symmetry](https://mlsanity.com/paper/2511.01064/) (2025)
- [The Within-Orbit Adaptive Leapfrog No-U-Turn Sampler](https://mlsanity.com/paper/2506.18746/) (2025)
- [Nearly Dimension-Independent Convergence of Mean-Field Black-Box Variational Inference](https://mlsanity.com/paper/2505.21721/) (2025)
- [Gaussian Mean Field Variational Inference can Overestimate Predictive Variance](https://mlsanity.com/paper/2606.25745/) (2026)
- [Divergence controls entropy in distillation](https://mlsanity.com/paper/2610.03529/) (2026)

## Abstract (arXiv)

When factorized approximations are used for variational inference (VI), they tend to underestimate the uncertainty -- as measured in various ways -- of the distributions they are meant to approximate. We consider two popular ways to measure the uncertainty deficit of VI: (i) the degree to which it underestimates the componentwise variance, and (ii) the degree to which it underestimates the entropy. To better understand these effects, and the relationship between them, we examine an informative setting where they can be explicitly (and elegantly) analyzed: the approximation of a Gaussian,~p, with a dense covariance matrix, by a Gaussian,~q, with a diagonal covariance matrix. We prove that q always underestimates both the componentwise variance and the entropy of p, textitthough not necessarily to the same degree. Moreover we demonstrate that the entropy of q is determined by the trade-off of two competing forces: it is decreased by the shrinkage of its componentwise variances (our first measure of uncertainty) but it is increased by the factorized approximation which delinks the nodes in the graphical model of p. We study various manifestations of this trade-off, notably one where, as the dimension of the problem grows, the per-component entropy gap between p and q becomes vanishingly small even though q underestimates every componentwise variance by a constant multiplicative factor. We also use the shrinkage-delinkage trade-off to bound the entropy gap in terms of the problem dimension and the condition number of the correlation matrix of p. Finally we present empirical results on both Gaussian and non-Gaussian targets, the former to validate our analysis and the latter to explore its limitations.