Sensitivity analysis in multilinear probabilistic models

Abstract

Sensitivity methods for the analysis of the outputs of discrete Bayesian networks have been extensively studied and implemented in different software packages. These methods usually focus on the study of sensitivity functions and on the impact of a parameter change to the Chan-Darwiche distance. Although not fully recognized, the majority of these results rely heavily on the multilinear structure of atomic probabilities in terms of the conditional probability parameters associated with this type of network. By defining a statistical model through the polynomial expression of its associated defining conditional probabilities, we develop here a unifying approach to sensitivity methods applicable to a large suite of models including extensions of Bayesian networks, for instance context-specific ones. Our algebraic approach enables us to prove that for models whose defining polynomial is multilinear both the Chan-Darwiche distance and any divergence in the family of φ-divergences are minimized for a certain class of multi-parameter contemporaneous variations when parameters are proportionally covaried.

Sensitivity analysis beyond linearity

“…In particular, it is proven that, for specific choices of parameters to be varied, proportional covariation is optimal, in the sense that it minimizes the CD distance between the original and varied distributions amongst all possible ways to covary parameters. Therefore, this work extends the results of Leonelli et al (2017a) for multilinear models to non-multilinear ones, as well as proposing sensitivity methods similar to those of Renooij (2012) and Charitos and van der Gaag (2006a) but which apply to a much more general class of models.…”

“…the exponents of the parameters are either zero or one. Leonelli et al (2017a) and Leonelli and Riccomagno (2018) give a thorough investigation of sensitivity analysis in multilinear MMs. Here conversely the focus is on models which are not necessarily multilinear.…”

“…where γ k and δ k need to be chosen so thatθ i + |θ S −i j | = 1 Different covariation schemes may entertain different properties which, depending on the domain of application, might be more or less desirable (see Leonelli et al, 2017a;Renooij, 2014, for a list). Applying a linear covariation scheme is very natural: if for instance δ k = −γ k , thenθ k = δ k (1 −θ i ) and the scheme assigns a proportion δ k of the remaining probability mass toθ k .…”

“…The measure of dissimilarity which is most commonly used in sensitivity analysis in graphical models is the so-called CD distance (Chan and Darwiche, 2005a). For single and specific multi-way parameter variations, proportional covariation minimizes the CD distance in BN models, as well as in any multilinear MM (Chan and Darwiche, 2002;Leonelli et al, 2017a). However, in nonmultilinear models even for single parameter variations proportional covariation does not minimize the CD distance in general as shown by the following example.…”

“…Most of the above-mentioned results, although specifically derived for BNs, hold for a variety of models whose atomic probabilities can be written as a multilinear polynomial (Leonelli et al, 2017a). The multilinear structure of atomic probabilities in BNs has been known for quite some time (Castillo et al, 1995;Darwiche, 2003), but other models entertain the same property under specific parametrisations, for instance stratified staged trees (Görgen et al, 2015), context-specific BNs (Boutilier et al, 1996) and influence diagrams (Leonelli et al, 2017b).…”