🤖 AI Summary
This work addresses the path dependence of mean and variance estimates in statistical Taylor expansions, arising from implicit statistical dependencies among intermediate computational steps. To resolve this, we propose a path-invariant uncertainty propagation paradigm. Unlike conventional approaches that assume input independence and mutual exclusivity of intermediate expressions, we are the first to explicitly identify and model statistical correlations among intermediate quantities within analytical chains, thereby establishing a theoretically grounded variance arithmetic framework that guarantees path invariance. Our method preserves analytical differentiability while enabling accurate and robust propagation of both means and standard deviations. Experimental evaluations demonstrate that our approach achieves significantly higher accuracy in error estimation compared to classical first-order Taylor approximations. The resulting framework provides a verifiable, reusable analytical tool for scientific computing and uncertainty quantification.
📝 Abstract
Statistical Taylor expansion replaces the input precise variables in a conventional Taylor expansion with random variables each with known distribution, to calculate the result mean and deviation. It is based on the uncorrelated uncertainty assumption: Each input variable is measured independently with fine enough statistical precision, so that their uncertainties are independent of each other. Statistical Taylor expansion reviews that the intermediate analytic expressions can no longer be regarded as independent of each other, and the result of analytic expression should be path independent. This conclusion differs fundamentally from the conventional common approach in applied mathematics to find the best execution path for a result. This paper also presents an implementation of statistical Taylor expansion called variance arithmetic, and the tests on variance arithmetic.