KL_bounds {bnmonitor} | R Documentation |
Computation of the bounds of the KL-divergence for variations of each parameter of a CI
object.
KL_bounds(ci, delta)
ci |
object of class |
delta |
multiplicative variation coefficient for the entry of the covariance matrix given in |
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.
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
.
C. GĂ¶rgen & M. Leonelli (2020), Model-preserving sensitivity analysis for families of Gaussian distributions. Journal of Machine Learning Research, 21: 1-32.
KL_bounds(synthetic_ci,1.05)