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 |

`delta` |
multiplicative variation coefficient for the entry of the covariance matrix given in |

### 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.

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

### See Also

### Examples

```
KL_bounds(synthetic_ci,1.05)
```

*bnmonitor*version 0.1.4 Index]