Computation of the bounds of the KL-divergence for variations of each parameter of a CI object.
Arguments
- ci
object of class
CI.- delta
multiplicative variation coefficient for the entry of the covariance matrix given in
entry.
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.
Details
Let \(\Sigma\) be the covariance matrix of a Gaussian Bayesian network with \(n\) vertices. Let \(D\) and \(\Delta\) be variation matrices acting additively on \(\Sigma\). Let also \(\tilde\Delta\) 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 $$KL(\tilde{Y}||Y)\leq 0.5n\max\left\{f(\lambda_{\max}(D\Sigma^{-1})),f(\lambda_{\min}(D\Sigma^{-1}))\right\}$$ whilst for a model-preserving one $$KL(\tilde{Y}||Y)\leq 0.5n\max\left\{f(\lambda_{\max}(\tilde\Delta\circ\Delta)),f(\lambda_{\min}(\tilde\Delta\circ\Delta))\right\}$$ where \(\lambda_{\max}\) and \(\lambda_{\min}\) 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.
References
C. Görgen & M. Leonelli (2020), Model-preserving sensitivity analysis for families of Gaussian distributions. Journal of Machine Learning Research, 21: 1-32.
Examples
KL_bounds(synthetic_ci,1.05)
#> row col standard total partial row_based col_based
#> 1 1 1 0.06319007 1.681933 1.622910 1.622910 1.622910
#> 2 1 2 0.71822823 1.681933 1.646925 1.630979 1.630979
#> 3 1 3 0.26097348 1.681933 1.646925 1.634914 1.634914
#> 4 1 4 NA 1.681933 NA NA NA
#> 5 2 2 0.26208872 1.681933 1.646925 1.630979 1.630979
#> 6 2 3 2.56494598 1.681933 1.646925 1.634914 1.634914
#> 7 2 4 NA 1.681933 NA NA NA
#> 8 3 3 0.63258146 1.681933 1.622910 1.622910 1.622910
#> 9 3 4 NA 1.681933 NA NA NA
#> 10 4 4 1.32814438 1.681933 1.622910 1.622910 1.622910