NAM_Entropy

Reproducible research code for computing and analyzing entropy-based metrics and related experiments. Built with Python 3.12, Poetry, and Jupyter Notebooks for a clean, portable workflow.

Table of Contents

• Overview
• Installation
• Project Structure
• Soft Entropy Estimates
• References

Overview

NAM_Entropy provides utilities and experiments for entropy-related analysis. Typical use cases include:

• Computing Shannon entropy, cross-entropy, KL-divergence, and related information-theoretic measures.
• Understanding the efficacy of sentence-level and token-level embeddings in a model by computing its information, variation, regularity, disentanglement.

Our target audience is:

• Cognitive Scientists who want an easy way to understand information-theoretic metrics for the information that a Nerual Network model carry about token and sentences via their embeddings.
• Maching Learning researchers who are interested in using parts of our efficient implementation in their workflow to understand the entropy and mutual information in their latent space model representations.
• Other researchers interested in computing entropy and information theory to their work.

Installation

To install the package dependencies, you can use either poetry or conda after changing to the package directory (containing pyproject.toml and environment.yml).

Install with Poetry

poetry install

(This creates a poetry virtual environment: nam-entropy-<hash>.)

Install with Conda

conda env create -f environment.yml

(This creates the conda virtual environment: nam-entropy-conda.)

Project Structure

nam_entropy/
├─ pyproject.toml
├─ poetry.lock
├─ README.md
├─ LICENSE
├─ .gitignore
├─ Makefile                           # optional convenience commands
├─ .env.example                       # example environment variables
├─ notebooks/
│  ├─ A. Automatic entropy and information calculations.ipynb
│  ├─ B. Interactive entropy and information calculations.ipynb
│  ├─ C. Programmatic entropy and information calculations.ipynb
│  ├─ D. Programmatic Spherical Entropy and Information Calculations.ipynb
│  └─ E. Online Spherical Entropy Calculations.ipynb
├─ src/
│  └─ nam_entropy/
│     ├─ __init__.py
│     ├─ bin_distribution_plots.py
│     ├─ data_prep.py
│     ├─ h.py
│     ├─ integrated_distribution_2d_sampler.py
│     ├─ make_data.py
│     └─ utils.py
└─ tests/

Soft Entropy Estimates

This package uses the soft entropy estimation methodology in Conklin (2025), Section 5.4 to provide a discrete approximation to differentiable entropy computation. This approach enables gradient-based optimization while maintaining the interpretability of traditional entropy measures. The key insight is mapping continuous neural representations to probability distributions over learned bin centers, preserving both representational nuances and computational tractability.

Metrics

1. Entropy

We compute the entropy $H(Z)$ of our given vector-valued distribution $Z$ by creating an associated soft-binned probability distrubtion $\mathrm{SoftBin}(Z)$ on a finite set of points randomly associated to the given dataset, and then compute the Shannon entropy $H(\mathrm{SoftBin}(Z))$ of this finite distribution.

For more details on Shannon entropy, see Wikipedia: Entropy or Science Direct: Shannon Entropy.

2. Conditional Entropy

We also can compute the conditional entropy of the given vector-valued distribution $Z$ with respect to a given discrete labelling of $Z$. We think of this as a (categorical) label random variable $L$ valued in the finite set of labels that shares a common probability space with $Z$. Then we compute the conditional entropy $H(Z|L)$ of $Z$ conditioned on (knowing) $L$ by the usual formula

$$H(Z|L) = \sum_{l \in L} p(l) H(Z|L=l)$$

where the entropy conditioned on the particular label value $L = l$ is given by

$$H(Z|L=l) = -\sum_{z \in Z} p(z|l) \log p(z|l),$$

and where

• $Z$ := Population data distribution (continuous vector-valued RV),
• $L$ := Sub-population label (categorical RV).

For more details on condtional entropy, see Wikipedia: Conditional Entropy or Science Direct: Conditional Entropy.

3. Mutual Information

The mutual information describes the amount of informatino that of two random variables give about each other. By definition, the mutual information $I(Z, L)$ of the two random variables $Z$ and $L$ is given by

$$I(Z, L) := H(Z) - H(Z|L),$$

where

• $Z$ := Population data distribution (continuous vector-valued RV),
• $L$ := Sub-population label (categorical RV).

For more details on mutual information, see Wikipedia: Mutual Information or Science Direct: Mutual Information.

References

Primary Theoretical Framework

Conklin, H. (2025). Information Structure in Mappings: An Approach to Learning, Representation, and Generalisation. PhD Thesis, University of Edinburgh. arXiv:2505.23960. https://arxiv.org/abs/2505.23960

For the mathematical foundations of soft entropy estimation used in this project, see Section 5.4: "Soft Entropy Estimation" (pp. 94-99), which provides the theoretical basis for the continuous-to-discrete entropy approximation methods implemented in our codebase.

Related Publications

Conklin, H., & Smith, K. (2024). Representations as Language: An Information-Theoretic Framework for Interpretability. International Meeting of the Cognitive Science Society. arXiv:2406.02449. https://arxiv.org/abs/2406.02449

Conklin, H., & Smith, K. (2024). Compositionality With Variation Reliably Emerges in Neural Networks. International Conference on Learning Representations (ICLR). https://openreview.net/forum?id=-Yzz6vlX7V-