# A robust estimator of mutual information for deep learning interpretability

arXiv: https://arxiv.org/abs/2211.00024 · HTML: https://mlsanity.com/paper/2211.00024/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 178 references that are in the map.
- First posted on arXiv 31 Oct 2022.

## Builds on (most cited first, 30 of 37)

- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Learning Deep Features for Discriminative Localization](https://mlsanity.com/paper/1512.04150/) (2015)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Learning deep representations by mutual information estimation and maximization](https://mlsanity.com/paper/1808.06670/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Grad-CAM++: Improved Visual Explanations for Deep Convolutional Networks](https://mlsanity.com/paper/1710.11063/) (2017)
- [Deep Variational Information Bottleneck](https://mlsanity.com/paper/1612.00410/) (2016)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [On Variational Bounds of Mutual Information](https://mlsanity.com/paper/1905.06922/) (2019)
- [The Variational Fair Autoencoder](https://mlsanity.com/paper/1511.00830/) (2015)
- [Understanding the Limitations of Variational Mutual Information Estimators](https://mlsanity.com/paper/1910.06222/) (2019)
- [Fader Networks: Manipulating Images by Sliding Attributes](https://mlsanity.com/paper/1706.00409/) (2017)
- [Interpretable Deep Learning: Interpretation, Interpretability, Trustworthiness, and Beyond](https://mlsanity.com/paper/2103.10689/) (2021)
- [Invariant Representations without Adversarial Training](https://mlsanity.com/paper/1805.09458/) (2018)
- [Variational Discriminator Bottleneck: Improving Imitation Learning, Inverse RL, and GANs by Constraining Information Flow](https://mlsanity.com/paper/1810.00821/) (2018)
- [Nonlinear Information Bottleneck](https://mlsanity.com/paper/1705.02436/) (2017)
- [A Review of Feature Selection Methods Based on Mutual Information](https://mlsanity.com/paper/1509.07577/) (2015)
- [Disentangling Factors of Variation with Cycle-Consistent Variational Auto-Encoders](https://mlsanity.com/paper/1804.10469/) (2018)
- [Efficient Estimation of Mutual Information for Strongly Dependent Variables](https://mlsanity.com/paper/1411.2003/) (2014)
- [A Survey of Deep Learning for Scientific Discovery](https://mlsanity.com/paper/2003.11755/) (2020)

## Built on by (most cited first, 4 of 4)

- [Explaining dark matter halo density profiles with neural networks](https://mlsanity.com/paper/2305.03077/) (2023)
- [A representation learning approach to probe for dynamical dark energy in matter power spectra](https://mlsanity.com/paper/2310.10717/) (2023)
- [LambdaCDM and early dark energy in latent space: a data-driven parametrization of the CMB temperature power spectrum](https://mlsanity.com/paper/2502.09810/) (2025)
- [Non-Stationarity Breaks Permutation Surrogates in Multi-Agent Reinforcement Learning: Diagnosis and Remedies](https://mlsanity.com/paper/2604.23716/) (2026)

## Abstract (arXiv)

We develop the use of mutual information (MI), a well-established metric in information theory, to interpret the inner workings of deep learning models. To accurately estimate MI from a finite number of samples, we present GMM-MI (pronounced ``Jimmie"), an algorithm based on Gaussian mixture models that can be applied to both discrete and continuous settings. GMM-MI is computationally efficient, robust to the choice of hyperparameters and provides the uncertainty on the MI estimate due to the finite sample size. We extensively validate GMM-MI on toy data for which the ground truth MI is known, comparing its performance against established mutual information estimators. We then demonstrate the use of our MI estimator in the context of representation learning, working with synthetic data and physical datasets describing highly non-linear processes. We train deep learning models to encode high-dimensional data within a meaningful compressed (latent) representation, and use GMM-MI to quantify both the level of disentanglement between the latent variables, and their association with relevant physical quantities, thus unlocking the interpretability of the latent representation. We make GMM-MI publicly available.