A geometric characterization of sensitivity analysis in monomial models

Abstract

Sensitivity analysis in probabilistic discrete graphical models is usually conducted by varying one probability at a time and observing how this affects output probabilities of interest. When one probability is varied, then others are proportionally covaried to respect the sum-to-one condition of probabilities. The choice of proportional covariation is justified by multiple optimality conditions, under which the original and the varied distributions are as close as possible under different measures. For variations of more than one parameter at a time and for the large class of discrete statistical models entertaining a regular monomial parametrisation, we demonstrate the optimality of newly defined proportional multi-way schemes with respect to an optimality criterion based on the I-divergence. We demonstrate that there are varying parameters' choices for which proportional covariation is not optimal and identify the sub-family of distributions where the distance between the original distribution and the one where probabilities are covaried proportionally is minimum. This is shown by adopting a new geometric characterization of sensitivity analysis in monomial models, which include most probabilistic graphical models. We also demonstrate the optimality of proportional covariation for multi-way analyses in Naive Bayes classifiers.