The Algebraic Limits of Polynomial Information Measures

2026-06-12

AI Summary

This work investigates the existence of polynomial dependence measures that satisfy the data processing inequality, vanish under independence, and admit unbiased estimators—serving as alternatives to mutual information. By analyzing the algebraic structure of symmetric polynomials on the probability simplex subject to information-theoretic constraints, the study establishes that no nontrivial solutions exist for asymmetric alphabets. In the symmetric setting, any feasible polynomial must contain the square of the determinant of the joint distribution matrix as a factor, yielding a degree lower bound of $2n$. These results delineate the algebraic infeasibility boundary for polynomial dependence measures and are applied to multi-task peer prediction mechanisms, providing a tight lower bound on the required number of tasks.

Abstract

The paper studies the existence of \emph{polynomial} measures of dependence between two random variables: polynomial functions of the joint distribution that (i) vanish on independence and (ii) cannot increase under post-processing of either variable (the data processing inequality, DPI). Mutual information satisfies both properties but is transcendental in the joint distribution, making it impossible to estimate without bias from finitely many samples. A polynomial alternative would admit an exact finite-sample unbiased estimator, with the polynomial degree controlling the required sample size. The main result is negative: in the asymmetric setting where the variables have different alphabet sizes (\(|X| > |Y|\)), no nonzero polynomial can simultaneously satisfy DPI on the larger-alphabet side and vanish on independence. In the symmetric case \(|X| = |Y| = n\), we establish a structural result: every such polynomial is divisible by \((\det U)^2\), where \(U\) denotes the \(n \times n\) joint distribution matrix. Consequently, any nontrivial candidate must have degree at least \(2n\). The determinant-based mutual information attains this lower bound. These algebraic results have direct consequences for \emph{multi-task peer prediction}, a mechanism-design problem in which a principal incentivizes honest reports from agents who observe correlated signals but no ground truth. Every such mechanism running on \(\ell\) independent tasks gives rise to a polynomial measure of dependence of degree at most \(\ell\), so our lower bounds on polynomial degree translate directly into lower bounds on the number of tasks: in the asymmetric case no finite-task mechanism exists on the larger-alphabet side at all, while the symmetric case requires at least \(\ell \geq 2n\) tasks.

Problem

• polynomial dependence measures
• data processing inequality
• mutual information
• finite-sample unbiased estimation
• multi-task peer prediction

Innovation

• polynomial dependence measures
• data processing inequality
• algebraic lower bounds
• multi-task peer prediction
• determinant-based mutual information