Partial information decomposition: redundancy as information bottleneck (2405.07665v2)

Abstract: The partial information decomposition (PID) aims to quantify the amount of redundant information that a set of sources provides about a target. Here, we show that this goal can be formulated as a type of information bottleneck (IB) problem, termed the "redundancy bottleneck" (RB). The RB formalizes a tradeoff between prediction and compression: it extracts information from the sources that best predict the target, without revealing which source provided the information. It can be understood as a generalization of "Blackwell redundancy", which we previously proposed as a principled measure of PID redundancy. The "RB curve" quantifies the prediction--compression tradeoff at multiple scales. This curve can also be quantified for individual sources, allowing subsets of redundant sources to be identified without combinatorial optimization. We provide an efficient iterative algorithm for computing the RB curve.

• The paper introduces the redundancy bottleneck, showing how to reframe Blackwell redundancy as an information bottleneck problem focused on shared data.
• It presents an iterative algorithm that efficiently computes the redundancy curve, enabling scalable analysis of high-dimensional datasets.
• The approach enhances multi-scale analysis and source identification, offering practical insights for fields like neuroscience and machine learning.

Exploring the Redundancy Bottleneck: A Novel Perspective on Information Theory

Introduction to the Concepts

The journey through understanding how information is shared among multiple sources in relation to a target variable has been a central theme in information theory, often explored through frameworks like Partial Information Decomposition (PID) and the Information Bottleneck (IB). Let's unpack the concepts gradually.

The Information Bottleneck (IB) Method

IB is a method designed for extracting relevant information from one variable (X) that is crucial for predicting another variable (Y). This method introduces a bottleneck variable (Q) which serves as a filtered form of X carrying only the necessary information needed about Y. The effectiveness of this filtration is measured by two terms:

1. Compression of X: Captured by mutual information between X and Q, $I(X;Q)$, indicating how much of X is squeezed into Q.
2. Prediction of Y: Described by the mutual information between Y and Q, $I(Y;Q)$, demonstrating how well Q predicts Y.

This set-up leads to a trade-off curve representing different balances of compression and prediction, helping customize information extraction based on specific needs.

Partial Information Decomposition (PID)

PID targets decomposing the information that a group of source variables provides about a target variable into components like redundancy and synergy. Redundancy reflects the shared information from all sources about the target, while synergy represents the unique predictive power arising from the collaboration of multiple sources beyond what they could accomplish individually.

Bridging PID with Information Bottleneck: The Redundancy Bottleneck

The concept of the Redundancy Bottleneck (RB) showcases a new bridging methodology between PID and IB, focusing initially on redundancy. Here’s how it functions and its implications:

Basic Formulation

The groundbreaking aspect of the Redundancy Bottleneck lies in its ability to remodel Blackwell redundancy into a format that resembles an IB problem but focuses on redundancy. By setting up a structure that trades off the amount of information (redundancy) used for prediction against the cost of compression or the specificity of information source, RB highlights scales of redundancy across sources.

Implications of RB

1. Multi-Scale Analysis: RB enables a finer examination of how redundancy varies across different predictive scales, offering a nuanced view of information distribution among sources.
2. Source Identification: With RB, it becomes practical to determine groups of sources that contribute most redundantly without combinatorial complications, a feature particularly handy in high-dimensional datasets.
3. Iterative Optimization: The authors elucidate an iterative algorithm to compute the RB curve efficiently, enhancing computational feasibility for larger and more complex datasets.

Theoretical and Practical Significance

The transition to viewing redundancy through the lens of the Information Bottleneck enriches theoretical understanding and adds practical tools for data scientists. From neuroscience to machine learning, understanding how redundancy scales with information compression can impact model interpretation and design, particularly in systems where redundancy might affect performance.

Future Directions

Looking ahead, the natural progression would be extending these ideas to other components like synergy or exploring how changes in the set-up of source variables might alter the observed redundancy. Additionally, adapting these theoretical constructs to real-world high-dimensional data could open new avenues in both research and application fields.

The Redundancy Bottleneck not only enriches the existing framework around information theory but also provides robust tools and concepts for tackling real-world data challenges where understanding intricate details of information flow and redundancy is crucial.