Fundamental Limits and Optimal Methods for Sharp Analytical Causal Bounds in Instrumental Variable Models

2026-04-14

AI Summary

This study addresses the computational challenge of efficiently deriving sharp analytical bounds on average treatment effects in discrete instrumental variable models. We establish a theoretical lower bound on computational complexity by proving that any sharp analytical bound must involve an exponential number of linear terms, and that the corresponding instrumental variable inequalities likewise grow exponentially in number. Leveraging tools from probability theory, linear programming, and combinatorial analysis, we develop an efficient constructive algorithm that matches this lower bound. We further provide open-source implementations in Python and R, which optimally generate sharp bounds and associated inequalities, thereby empirically validating our theoretical results.

Technology Category

• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Bounding causal effects analytically, rather than numerically, is appealing for its interpretability and conceptual clarity. Existing sharp methods rely on optimization-based approaches such as the Balke-Pearl framework, whose computational complexity grows rapidly. An alternative line of work derives bounds heuristically using probability laws and generic inequalities, and some recent papers have claimed or conjectured that this approach can yield sharp analytical bounds with substantially lower complexity. In this paper, we show that this perceived advantage is illusory. In particular, in a discrete instrumental variable setting, we show that any sharp analytical bound for the average treatment effect must be expressible as a maximum (minimum) over a collection of linear terms whose cardinality grows exponentially in the number of values taken by the outcome. In parallel, we show that the number of instrumental variable inequalities itself also grows exponentially. Consequently, bounds and inequalities expressed using only polynomially many such terms cannot be sharp. As a constructive complement, the paper is accompanied by codes implemented in python and R to derive sharp analytical bounds and sharp inequalities with optimal efficiency, matching the lower bounds proven in this paper. These codes are available online.

Problem

• instrumental variable
• causal bounds
• analytical bounds
• computational complexity
• sharpness

Innovation

• instrumental variable
• causal bounds
• analytical sharpness
• computational complexity
• exponential lower bound