  Sensitivity analysis in probabilistic discrete graphical models is usually
conducted by varying one probability value 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
probability laws. The choice of proportional covariation is justified by a
variety of optimality conditions, under which the original and the varied
distributions are as close as possible under different measures of closeness.
For variations of more than one parameter at a time proportional covariation is
justified in some special cases only. In this work, 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 notion of I-divergence. We
demonstrate that there are varying parameters choices for which proportional
covariation is not optimal and identify the sub-family of model 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 formal, geometric characterization of sensitivity analysis in monomial
models, which include a wide array of probabilistic graphical models. We also
demonstrate the optimality of proportional covariation for multi-way analyses
in Naive Bayes classifiers.
