An information upper bound for probability sensitivity

TL;DR

This paper establishes a new mathematical upper bound for probability sensitivity in models with uncertain inputs, using information-theoretic metrics like Fisher information and Kullback-Leibler divergence, aiding decision-making.

Contribution

It provides the first known probability sensitivity bound based on information-theoretic metrics, proven using elementary inequalities, and extends this bound to Fisher information of inputs and outputs.

Findings

The probability sensitivity is bounded by information-theoretic metrics.

The proof relies on Titu's lemma, an elementary inequality.

Numerical examples illustrate the bounds' applicability.

Abstract

Uncertain input of a mathematical model induces uncertainties in the output and probabilistic sensitivity analysis identifies the influential inputs to guide decision-making. Of practical concern is the probability that the output would, or would not, exceed a threshold, and the probability sensitivity depends on this threshold which is often uncertain. The Fisher information and the Kullback-Leibler divergence have been recently proposed in the literature as threshold-independent sensitivity metrics. We present mathematical proof that the information-theoretical metrics provide an upper bound for the probability sensitivity. The proof is elementary, relying only on a special version of the Cauchy-Schwarz inequality called Titu's lemma. Despite various inequalities exist for probabilities, little is known of probability sensitivity bounds and the one proposed here is new to the present authors' knowledge. The probability sensitivity bound is extended, analytically and with numerical examples, to the Fisher information of both the input and output. It thus provides a solid mathematical basis for decision-making based on probabilistic sensitivity metrics.