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

arXiv: https://arxiv.org/abs/2505.12094 · HTML: https://mlsanity.com/paper/2505.12094/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 15 references that are in the map.
- First posted on arXiv 17 May 2025.

## Builds on (most cited first, 5 of 5)

- [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)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Causal Shapley Values: Exploiting Causal Knowledge to Explain Individual Predictions of Complex Models](https://mlsanity.com/paper/2011.01625/) (2020)
- [The Blessings of Multiple Causes](https://mlsanity.com/paper/1805.06826/) (2018)

## Abstract (arXiv)

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.