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Attribution Projection Calculus: A Novel Framework for Causal Inference in Bayesian Networks

M Ruhul Amin

TL;DR

AP-Calculus introduces a structured Bayesian network with a source, intermediate deconfounding nodes, and a destination to achieve label-specific causal attribution. It formalizes deconfounder and confounder roles, defines separation functions, and proves architectural optimality for causal inference relative to alternatives including Pearl's do-calculus, while providing extensions for attribution, fairness, information gain, and uncertainty analysis. The framework includes rigorous theoretical guarantees, dimensional sufficiency for intermediate representations, and practical extensions to large language models and real-world tasks. Its combination of principled causal reasoning, scalable attribution, and applicability to high-dimensional models offers a robust path toward interpretable, fair, and reliable AI systems.

Abstract

This paper introduces Attribution Projection Calculus (AP-Calculus), a novel mathematical framework for determining causal relationships in structured Bayesian networks. We investigate a specific network architecture with source nodes connected to destination nodes through intermediate nodes, where each input maps to a single label with maximum marginal probability. We prove that for each label, exactly one intermediate node acts as a deconfounder while others serve as confounders, enabling optimal attribution of features to their corresponding labels. The framework formalizes the dual nature of intermediate nodes as both confounders and deconfounders depending on the context, and establishes separation functions that maximize distinctions between intermediate representations. We demonstrate that the proposed network architecture is optimal for causal inference compared to alternative structures, including those based on Pearl's causal framework. AP-Calculus provides a comprehensive mathematical foundation for analyzing feature-label attributions, managing spurious correlations, quantifying information gain, ensuring fairness, and evaluating uncertainty in prediction models, including large language models. Theoretical verification shows that AP-Calculus not only extends but can also subsume traditional do-calculus for many practical applications, offering a more direct approach to causal inference in supervised learning contexts.

Attribution Projection Calculus: A Novel Framework for Causal Inference in Bayesian Networks

TL;DR

AP-Calculus introduces a structured Bayesian network with a source, intermediate deconfounding nodes, and a destination to achieve label-specific causal attribution. It formalizes deconfounder and confounder roles, defines separation functions, and proves architectural optimality for causal inference relative to alternatives including Pearl's do-calculus, while providing extensions for attribution, fairness, information gain, and uncertainty analysis. The framework includes rigorous theoretical guarantees, dimensional sufficiency for intermediate representations, and practical extensions to large language models and real-world tasks. Its combination of principled causal reasoning, scalable attribution, and applicability to high-dimensional models offers a robust path toward interpretable, fair, and reliable AI systems.

Abstract

This paper introduces Attribution Projection Calculus (AP-Calculus), a novel mathematical framework for determining causal relationships in structured Bayesian networks. We investigate a specific network architecture with source nodes connected to destination nodes through intermediate nodes, where each input maps to a single label with maximum marginal probability. We prove that for each label, exactly one intermediate node acts as a deconfounder while others serve as confounders, enabling optimal attribution of features to their corresponding labels. The framework formalizes the dual nature of intermediate nodes as both confounders and deconfounders depending on the context, and establishes separation functions that maximize distinctions between intermediate representations. We demonstrate that the proposed network architecture is optimal for causal inference compared to alternative structures, including those based on Pearl's causal framework. AP-Calculus provides a comprehensive mathematical foundation for analyzing feature-label attributions, managing spurious correlations, quantifying information gain, ensuring fairness, and evaluating uncertainty in prediction models, including large language models. Theoretical verification shows that AP-Calculus not only extends but can also subsume traditional do-calculus for many practical applications, offering a more direct approach to causal inference in supervised learning contexts.
Paper Structure (64 sections, 20 theorems, 61 equations, 5 figures, 1 table, 3 algorithms)

This paper contains 64 sections, 20 theorems, 61 equations, 5 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Causal Inference and Do-Calculus
  4. Bayesian Networks for Causal Inference
  5. Confounding and Deconfounding
  6. Feature Attribution Methods
  7. Network Architecture and Problem Formulation
  8. Bayesian Network Structure
  9. Problem Formulation
  10. Mathematical Representation
  11. Attribution Projection Calculus
  12. Core Definitions
  13. Fundamental Axioms of AP-Calculus
  14. Deconfounder Analysis
  15. Dimensional Analysis of Intermediate Nodes
  16. ...and 49 more sections

Key Result

Theorem 1

For a given input tuple $\mathbf{t}$ and label $l$, considering $x_l$ as the deconfounder and all other intermediate nodes as confounders maximizes the marginal probability of label $l$:

Figures (5)

  • Figure 1: The proposed Bayesian network architecture with source node $s$, destination node $d$, and intermediate nodes $\{x_1, x_2, \ldots, x_m\}$.
  • Figure 2: The proposed Bayesian network architecture with source node $s$, destination node $d$, and intermediate nodes $\{x_1, x_2, \ldots, x_m\}$.
  • Figure 3: Pearl's junction structure with common causes $a$ and $b$ affecting both source $s$ and destination $d$.
  • Figure 4: Common cause structure where $s$ is a common cause of $d$, $a$, and $b$.
  • Figure 5: The network architecture with $x_1$ as a deconfounder for label 1 and other nodes as confounders.

Theorems & Definitions (47)

  • Definition 4.1: Deconfounder Node
  • Definition 4.2: Confounder Node
  • Definition 4.3: Separation Function
  • Definition 4.4: Attribution Projection
  • Theorem 1: Deconfounder Optimality
  • proof
  • Theorem 2: Dimensional Sufficiency
  • proof
  • Theorem 3: Separation Function Optimality
  • proof
  • ...and 37 more