# Sensitivity analysis beyond linearity

## TL;DR

This paper extends sensitivity analysis methods to non-multilinear probabilistic graphical models, showing that sensitivity functions are polynomial and identifying optimal covariation schemes.

## Contribution

It introduces a general approach for sensitivity analysis in non-multilinear models, relaxing the multilinearity assumption and deriving polynomial sensitivity functions.

## Key findings

- Sensitivity functions are polynomial in non-multilinear models.
- Proportional covariation is optimal under certain conditions.
- Derived divergence and distance measures for various covariation schemes.

## Abstract

A wide array of graphical models can be parametrised to have atomic probabilities represented by monomial functions. Such monomial structure has proven very useful when studying robustness under the assumption of a multilinear model where all monomial have either zero or one exponents. Robustness in probabilistic graphical models is usually investigated by varying some of the input probabilities and observing the effects of these on output probabilities of interest. Here the assumption of multilinearity is relaxed and a general approach for sensitivity analysis in non-multilinear models is presented. It is shown that in non-multilinear models sensitivity functions have a polynomial form, conversely to multilinear models where these are simply linear. The form of various divergences and distances under different covariation schemes is also formally derived. Proportional covariation is proven to be optimal in non-multilinear models under some specific choices of varied parameters. The methodology is illustrated throughout by an educational application.