# A Practical & Unified Notation for Information-Theoretic Quantities in ML

arXiv: https://arxiv.org/abs/2106.12062 · HTML: https://mlsanity.com/paper/2106.12062/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 29 references that are in the map.
- First posted on arXiv 22 Jun 2021.

## Builds on (most cited first, 9 of 9)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Deep Bayesian Active Learning with Image Data](https://mlsanity.com/paper/1703.02910/) (2017)
- [Unifying Count-Based Exploration and Intrinsic Motivation](https://mlsanity.com/paper/1606.01868/) (2016)
- [Opening the Black Box of Deep Neural Networks via Information](https://mlsanity.com/paper/1703.00810/) (2017)
- [Variational Dropout and the Local Reparameterization Trick](https://mlsanity.com/paper/1506.02557/) (2015)
- [BatchBALD: Efficient and Diverse Batch Acquisition for Deep Bayesian Active Learning](https://mlsanity.com/paper/1906.08158/) (2019)
- [A Theory of Usable Information Under Computational Constraints](https://mlsanity.com/paper/2002.10689/) (2020)
- [Variational Bayesian Optimal Experimental Design](https://mlsanity.com/paper/1903.05480/) (2019)
- [Unpacking Information Bottlenecks: Unifying Information-Theoretic Objectives in Deep Learning](https://mlsanity.com/paper/2003.12537/) (2020)

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

- [Unifying Approaches in Active Learning and Active Sampling via Fisher Information and Information-Theoretic Quantities](https://mlsanity.com/paper/2208.00549/) (2022)
- [Navigating the Pitfalls of Active Learning Evaluation: A Systematic Framework for Meaningful Performance Assessment](https://mlsanity.com/paper/2301.10625/) (2023)
- [A Note on "Assessing Generalization of SGD via Disagreement"](https://mlsanity.com/paper/2202.01851/) (2022)
- [Marginal and Joint Cross-Entropies & Predictives for Online Bayesian Inference, Active Learning, and Active Sampling](https://mlsanity.com/paper/2205.08766/) (2022)

## Abstract (arXiv)

A practical notation can convey valuable intuitions and concisely express new ideas. Information theory is of importance to machine learning, but the notation for information-theoretic quantities is sometimes opaque. We propose a practical and unified notation and extend it to include information-theoretic quantities between observed outcomes (events) and random variables. This includes the point-wise mutual information known in NLP and mixed quantities such as specific surprise and specific information in the cognitive sciences and information gain in Bayesian optimal experimental design. We apply our notation to prove a version of Stirling's approximation for binomial coefficients mentioned by MacKa (2003) using new intuitions. We also concisely rederive the evidence lower bound for variational auto-encoders and variational inference in approximate Bayesian neural networks. Furthermore, we apply the notation to a popular information-theoretic acquisition function in Bayesian active learning which selects the most informative (unlabelled) samples to be labelled by an expert and extend this acquisition function to the core-set problem with the goal of selecting the most informative samples given the labels.