KL_bounds {bnmonitor} R Documentation

Bounds for the KL-divergence

Description

Computation of the bounds of the KL-divergence for variations of each parameter of a CI object.

Usage

KL_bounds(ci, delta)

Arguments

 ci object of class CI. delta multiplicative variation coefficient for the entry of the covariance matrix given in entry.

Details

Let Σ be the covariance matrix of a Gaussian Bayesian network with n vertices. Let D and Δ be variation matrices acting additively on Σ. Let also \tildeΔ be a model-preserving co-variation matrix. Denote with Y and \tilde{Y} the original and the perturbed random vectors. Then for a standard sensitivity analysis

KL(\tilde{Y}||Y)≤q 0.5n\max≤ft\{f(λ_{\max}(DΣ^{-1})),f(λ_{\min}(DΣ^{-1}))\right\}

whilst for a model-preserving one

KL(\tilde{Y}||Y)≤q 0.5n\max≤ft\{f(λ_{\max}(\tildeΔ\circΔ)),f(λ_{\min}(\tildeΔ\circΔ))\right\}

where λ_{\max} and λ_{\min} are the largest and the smallest eigenvalues, respectively, f(x)=\ln(1+x)-x/(1+x) and \circ denotes the Schur or element-wise product.

Value

A dataframe including the KL-divergence bound for each co-variation scheme (model-preserving and standard) and every entry of the covariance matrix. For variations leading to non-positive semidefinite matrix, the dataframe includes a NA.

References

C. Görgen & M. Leonelli (2020), Model-preserving sensitivity analysis for families of Gaussian distributions. Journal of Machine Learning Research, 21: 1-32.