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Description

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CD-distance

Description

Chan-Darwiche (CD) distance between a Bayesian network and its update after parameter variation.

Usage

CD(
  bnfit,
  node,
  value_node,
  value_parents,
  new_value,
  covariation = "proportional"
)

Arguments

Details

The Bayesian network on which parameter variation is being conducted should be expressed as a bn.fit object. The name of the node to be varied, its level and its parent's levels should be specified. The parameter variation specified by the function is:

P ( node = value_node | parents = value_parents ) = new_value

The CD distance between two probability distributions P and P' defined over the same sample space \mathcal{Y} is defined as

CD(P,P')= \log\max_{y\in\mathcal{Y}}\left(\frac{P(y)}{P'(y)}\right) - \log\min_{y\in\mathcal{Y}}\left(\frac{P(y)}{P'(y)}\right)

Value

The function CD returns a dataframe including in the first column the variations performed, and in the following columns the corresponding CD distances for the chosen co-variation schemes.

References

Chan, H., & Darwiche, A. (2005). A distance measure for bounding probabilistic belief change. International Journal of Approximate Reasoning, 38(2), 149-174.

Renooij, S. (2014). Co-variation for sensitivity analysis in Bayesian networks: Properties, consequences and alternatives. International Journal of Approximate Reasoning, 55(4), 1022-1042.

See Also

KL.bn.fit

Examples

CD(synthetic_bn, "y2", "1", "2", "all", "all")
CD(synthetic_bn, "y1", "2", NULL, 0.3, "all")

Frobenius norm

Description

Fro returns the Frobenius norm between a Bayesian network and its update after parameter variation.

Usage

Fro(x, ...)

Arguments

Details

The details depend on the class the method Fro is applied to.

Value

A dataframe whose columns depend of the class of the object.

See Also

KL.GBN, KL.CI, Fro.CI, Fro.GBN, Jeffreys.GBN, Jeffreys.CI

Frobenius norm for CI

Description

Fro.CI returns the Frobenius norm between an object of class CI and its update after a model-preserving parameter variation.

Usage

## S3 method for class 'CI'
Fro(x, type, entry, delta, log = TRUE, ...)

Arguments

Details

Computation of the Frobenius norm between a Bayesian network and its updated version after a model-preserving variation.

Value

A dataframe including in the first column the variations performed, and in the following columns the corresponding Frobenius norms for the chosen model-preserving co-variations.

References

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

See Also

KL.GBN, KL.CI, Fro.GBN, Jeffreys.GBN, Jeffreys.CI

Examples

Fro(synthetic_ci,"total",c(1,1),seq(0.9,1.1,0.01))
Fro(synthetic_ci,"partial",c(1,4),seq(0.9,1.1,0.01))
Fro(synthetic_ci,"column",c(1,2),seq(0.9,1.1,0.01))
Fro(synthetic_ci,"row",c(3,2),seq(0.9,1.1,0.01))

Frobenius norm for GBN

Description

Fro.GBN returns the Frobenius norm between between an object of class GBN and its update after a standard parameter variation.

Usage

## S3 method for class 'GBN'
Fro(x, entry, delta, log = TRUE, ...)

Arguments

Details

Computation of the Frobenius norm between a Bayesian network and the additively perturbed Bayesian network, where the perturbation is either to the mean vector or to the covariance matrix. The Frobenius norm is not computed for perturbations of the mean since it is always equal to zero.

Value

A dataframe including in the first column the variations performed and in the second column the corresponding Frobenius norm.

See Also

KL.GBN, KL.CI, Fro.CI, Jeffreys.GBN, Jeffreys.CI

Examples

Fro(synthetic_gbn,c(3,3),seq(-1,1,0.1))

Jeffreys Divergence

Description

Jeffreys returns the Jeffreys divergence between a continuous Bayesian network and its update after parameter variation.

Usage

Jeffreys(x, ...)

Arguments

Details

The details depend on the class the method Jefrreys is applied to.

Value

A dataframe whose columns depend of the class of the object.

See Also

KL.GBN, KL.CI, Fro.CI, Fro.GBN, Jeffreys.GBN, Jeffreys.CI

Jeffreys Divergence for CI

Description

Jeffreys.CI returns the Jeffreys divergence between an object of class CI and its update after a model-preserving parameter variation.

Usage

## S3 method for class 'CI'
Jeffreys(x, type, entry, delta, ...)

Arguments

Details

Computation of the Jeffreys divergence between a Bayesian network and its updated version after a model-preserving variation.

Value

A dataframe including in the first column the variations performed, and in the following columns the corresponding Jeffreys divergences for the chosen model-preserving co-variations.

References

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

See Also

KL.GBN, KL.CI, Fro.CI, Fro.GBN, Jeffreys.GBN

Examples

Jeffreys(synthetic_ci,"total",c(1,1),seq(0.9,1.1,0.01))
Jeffreys(synthetic_ci,"partial",c(1,4),seq(0.9,1.1,0.01))
Jeffreys(synthetic_ci,"column",c(1,2),seq(0.9,1.1,0.01))
Jeffreys(synthetic_ci,"row",c(3,2),seq(0.9,1.1,0.01))
Jeffreys(synthetic_ci,"all",c(3,2),seq(0.9,1.1,0.01))

Jeffreys Divergence for GBN

Description

Jeffreys.GBN returns the Jeffreys divergence between an object of class GBN and its update after a standard parameter variation.

Usage

## S3 method for class 'GBN'
Jeffreys(x, where, entry, delta, ...)

Arguments

Details

Computation of the Jeffreys divergence between a Bayesian network and the additively perturbed Bayesian network, where the perturbation is either to the mean vector or to the covariance matrix.

Value

A dataframe including in the first column the variations performed and in the second column the corresponding Jeffreys divergences.

References

Goergen, C., & Leonelli, M. (2018). Model-preserving sensitivity analysis for families of Gaussian distributions. arXiv preprint arXiv:1809.10794.

See Also

KL.GBNKL.CI, Fro.CI, Fro.GBN, Jeffreys.CI

Examples

Jeffreys(synthetic_gbn,"mean",2,seq(-1,1,0.1))
Jeffreys(synthetic_gbn,"covariance",c(3,3),seq(-1,1,0.1))

KL Divergence

Description

KL returns the Kullback-Leibler (KL) divergence between a Bayesian network and its update after parameter variation.

Usage

KL(x, ...)

Arguments

Details

The details depend on the class the method KL is applied to.

Value

A dataframe whose columns depend of the class of the object.

See Also

KL.GBN, KL.CI, Fro.CI, Fro.GBN, Jeffreys.GBN, Jeffreys.CI

KL Divergence for CI

Description

KL.CI returns the Kullback-Leibler (KL) divergence between an object of class CI and its update after a model-preserving parameter variation.

Usage

## S3 method for class 'CI'
KL(x, type, entry, delta, ...)

Arguments

Details

Computation of the KL divergence between a Bayesian network and its updated version after a model-preserving variation.

Value

A dataframe including in the first column the variations performed, and in the following columns the corresponding KL divergences for the chosen model-preserving co-variations.

References

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

See Also

KL.GBN, Fro.CI, Fro.GBN, Jeffreys.GBN, Jeffreys.CI

Examples

KL(synthetic_ci, "total", c(1,1), seq(0.9,1.1,0.01))
KL(synthetic_ci, "partial", c(1,4), seq(0.9,1.1,0.01))
KL(synthetic_ci, "column", c(1,2), seq(0.9,1.1,0.01))
KL(synthetic_ci, "row", c(3,2), seq(0.9,1.1,0.01))
KL(synthetic_ci, "all", c(3,2), seq(0.9,1.1,0.01))

KL Divergence for GBN

Description

KL.GBN returns the Kullback-Leibler (KL) divergence between an object of class GBN and its update after a standard parameter variation.

Usage

## S3 method for class 'GBN'
KL(x, where, entry, delta, ...)

Arguments

Details

Computation of the KL divergence between a Bayesian network and the additively perturbed Bayesian network, where the perturbation is either to the mean vector or to the covariance matrix.

Value

A dataframe including in the first column the variations performed and in the second column the corresponding KL divergences.

References

Gómez-Villegas, M. A., Maín, P., & Susi, R. (2007). Sensitivity analysis in Gaussian Bayesian networks using a divergence measure. Communications in Statistics—Theory and Methods, 36(3), 523-539.

Gómez-Villegas, M. A., Main, P., & Susi, R. (2013). The effect of block parameter perturbations in Gaussian Bayesian networks: Sensitivity and robustness. Information Sciences, 222, 439-458.

See Also

KL.CI, Fro.CI, Fro.GBN, Jeffreys.GBN, Jeffreys.CI

Examples

KL(synthetic_gbn,"mean",2,seq(-1,1,0.1))
KL(synthetic_gbn,"covariance",c(3,3),seq(-1,1,0.1))

KL Divergence for bn.fit

Description

KL.bn.fit returns the Kullback-Leibler (KL) divergence between a Bayesian network and its update after parameter variation.

Usage

## S3 method for class 'bn.fit'
KL(
  x,
  node,
  value_node,
  value_parents,
  new_value,
  covariation = "proportional",
  ...
)

Arguments

Details

The Bayesian network on which parameter variation is being conducted should be expressed as a bn.fit object. The name of the node to be varied, its level and its parent's levels should be specified. The parameter variation specified by the function is:

P ( node = value_node | parents = value_parents ) = new_value

Value

A dataframe with the varied parameter and the KL divergence for different co-variation schemes. If plot = TRUE the function returns a plot of the KL divergences.

References

Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. The annals of mathematical statistics, 22(1), 79-86.

Leonelli, M., Goergen, C., & Smith, J. Q. (2017). Sensitivity analysis in multilinear probabilistic models. Information Sciences, 411, 84-97.

See Also

CD

Examples

KL(synthetic_bn, "y2", "1", "2", "all", "all")
KL(synthetic_bn, "y1", "2", NULL, 0.3, "all")

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

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

See Also

KL.CI, KL.CI

Examples

KL_bounds(synthetic_ci,1.05)

Amalgamation of levels

Description

Computation of the diameter of children conditional probability tables when levels of a variable are merged.

Usage

amalgamation(bnfit, node)

Arguments

Value

A list where each entry refers to a child of node. For each child, the function reports the diameter resulting from the amalgamation of any pair of levels.

References

Leonelli, M., Smith, J. Q., & Wright, S. K. (2024). The diameter of a stochastic matrix: A new measure for sensitivity analysis in Bayesian networks. arXiv preprint arXiv:2407.04667.

Examples

amalgamation(synthetic_bn, "y1")

Measures of asymmetric independence

Description

Computation of the indexes of context-specific and partial independence in a conditional probability table.

Usage

asy_measure(bnfit, node)

Arguments

Details

The index of context-specific independence computes the upper diameter of a CPT where all parents but one are fixed, while the index of partial independence computes the lower diameter for the same CPT. If the lower diameter is close to zero it means that there are at least two rows of the CPT which are very similar to each other, thus implying a partial conditional independence.

Value

A list where each entry refers to one of the parent of node. Each entry of the list is a dataframe with the index of context-specific independence for each outcome of the conditioning parent in one column and in additional column the index of partial independence in the case of non-binary variables.

References

Leonelli, M., Smith, J. Q., & Wright, S. K. (2024). The diameter of a stochastic matrix: A new measure for sensitivity analysis in Bayesian networks. arXiv preprint arXiv:2407.04667.

Pensar, J., Nyman, H., Lintusaari, J., & Corander, J. (2016). The role of local partial independence in learning of Bayesian networks. International Journal of Approximate Reasoning, 69, 91-105.

See Also

diameter

Examples

asy_measure(synthetic_bn, "y3")

Integration with bn.fit objects from bnlearn

Description

Functions that transform an object of class bn.fit and bn.fit.gnet (a Gaussian Bayesian network) to objects of class GBN or CI.

Usage

bn2gbn(bnfit)

bn2ci(bnfit)

Arguments

Value

The function bn2gbn returns an object of class GBN consisting of a list with entries:

• order: An ordering of the nodes according to the graph.

• mean: The mean vector of the Gaussian distribution.

• covariance: The covariance matrix of the Gaussian distribution.

The function bn2ci returns an object of class CI consisting of the same list as GBN, but with the additional entry cond_ind. cond_ind is a list where each entry consists of A, B and C corresponding to the conditional independence statements A independent of B given C embedded by the network.

bnmonitor: A package for sensitivity analysis and robustness in Bayesian networks

Description

Sensitivity and robustness analysis for Bayesian networks.

Details

bnmonitor provides functions to perform sensitivity analysis for both discrete Bayesian networks (DBNs) and Gaussian Bayesian networks (GBNs). The following types of sensitivity investigations are available:

• Parametric sensitivity analysis: Investigate the effect of changes in some of the parameter values in a Bayesian network and quantify the difference between the original and perturbed Bayesian networks using dissimilarity measures (both for DBNs and GBNs).

• Robustness to data: Verify how well a Bayesian network fits a specific dataset that was used either for learning or for testing (only for DBNs).

• Node influence: Quantify how much the nodes of a Bayesian network influence an output node of interest (only for DBNs).

• Edge strength: Assess the strength of the edges of a Bayesian network (only for DBNs).

• Other investigations: Including the diameter of the conditional probability tables, measures of asymmetric independence, and level amalgamation.

DBNs - Robustness to data

The available functions for robustness are:

• Node monitors (node_monitor): contribution of each vertex to the overall log-likelihood of the model.

• Observation's influence (influential_obs): difference in the log-likelihood of a model learnt with the full dataset and with all but one observation.

• Final node monitors (node_monitor): marginal and conditional node monitors to assess the fit of a vertex distribution to the full dataset.

• Sequential node monitors (seq_node_monitor): marginal and conditional node monitors for a specific vertex only using sequentially subsets of the dataset.

• Sequential parent-child monitor (seq_pa_ch_monitor): parent-child node monitor for a specific vertex and a specific configuration of its parents using sequentially subsets of the dataset.

DBNs - Co-variation schemes

The available co-variation schemes are:

• Uniform co-variation scheme (uniform_covar): distributes the probability mass to be co-varied uniformly among the co-varying parameters.

• Proportional co-variation scheme (proportional_covar): distributes the probability mass to be co-varied in the same proportion as in the original Bayesian network.

• Order-preserving co-variation scheme (orderp_covar):distributes the to be co-varied probability mass among the co-varying parameters so that the original order of parameters is preserved.

DBNs - Dissimilarity measures

The dissimilarity measures quantify the difference between a Bayesian network and its update after parameter variation.
The available dissimilarity measures are:

• Chan-Darwiche distance (CD)

• Kullback-Leibler divergence (KL)

DBNs - Sensitivity functions

The available functions for sensitivity analysis are:

• Sensitivity function (sensitivity): returns a certain probability of interest given a parameter change. Evidence can be considered.

• Sensitivity query (sensquery): returns the parameter changes needed to get a certain probability of interest. Evidence can be considered.

DBNs - Node influence

The available functions for node influence are:

• Mutual information (mutual_info): returns the mutual information between a node and all other nodes of a DBN.

• Distance-weighted influence (dwi): returns the distance-weighted influence between a node and all other nodes of a DBN.

• Edge-weighted influence (ewi): returns the edge-weighted influence between a node and all other nodes of a DBN.

DBNs - Edge strength

The available functions for edge strength are:

• Measure of edge strength (edge_strength): returns the edge strength measure for all edges of a DBN.

DBNs - Other sensitivity measures

Other sensitivity measures available are:

• Diameter (diameter): returns the diameter of the conditional probability tables of all non-root nodes of a DBN.

• Level amalgamation (amalgamation): returns the diameter of all children conditional probability tables of a node in DBN when every pair of levels are merged.

• Measures of asymmetric independence (asy_measure): returns the indexes of context-specific and partial conditional independence for all variables of a DBN.

GBNs - Model-Preserving matrices

The available functions to construct model-preserving co-variation matrices are:

• Total co-variation matrix (total_covar_matrix).

• Partial co-variation matrix (partial_covar_matrix).

• Row-based co-variation matrix (row_covar_matrix).

• Column-based co-variation matrix (col_covar_matrix).

GBNs - Mean and Covariance variations

The available functions to perturb the distribution of a GBN are:

• Mean variations (mean_var).

• Standard covariance variations (covariance_var).

• Model-preserving covariance variations (model_pres_cov).

GBNs - Dissimilarity measures

The available dissimilarity measures are:

• Frobenius norm (Fro).

• Jeffrey's distance (Jeffreys).

• Kullback-Leibler divergence (KL).

• Upper bound to the KL divergence (KL_bounds).

Bayesian networks for a cachexia study

Description

Continuous Bayesian networks comparing the dependence of metabolomics for people who suffer and do not suffer of Cachexia

Usage

cachexia_gbn

cachexia_ci

control_gbn

control_ci

cachexia_data

Format

Continuous Bayesian networks over six metabolomics: Adipate (A), Betaine (B), Fumarate (F), Glucose (GC), Glutamine (GM) and Valine (V). The networks cachexia_gbn and cachexia_ci are for people suffering of cachexia and of class GBN and CI respectively. The networks control_gbn and control_ci are for people not suffering of cachexia and of class GBN and CI respectively. The original dataset is stored in cachexia_data.

An object of class GBN of length 3.

An object of class CI of length 4.

An object of class GBN of length 3.

An object of class CI of length 4.

An object of class data.table (inherits from data.frame) with 77 rows and 7 columns.

References

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

Christchurch Health and Development Study

Description

Simulated data and Bayesian networks from the Christchurch Health and Development Study

Usage

chds

chds_bn

chds_bn.fit

Format

The dataframe chds includes 500 observations randomly simulated from the bn.fit object chds_bn.fit. It has four variables:

• Social: family's social background with levels "High" and "Low"

• Economic: family's economic status with levels "High" and "Low"

• Events: number of family life events with levels "High", "Average" and "Low"

• Admission: hospital admission of the child with levels "yes" and "no"

• statistics: mark out of 100 for statistics

chds_bn is an object of class bn including the MAP Bayesian network from Barclay et al. (2013) and chds_bn.fit is an object of class bn.fit including the probabilities from the same article.

An object of class data.frame with 500 rows and 4 columns.

An object of class bn of length 3.

An object of class bn.fit (inherits from bn.fit.dnet) of length 4.

References

Fergusson, D. M., Horwood, L. J., & Shannon, F. T. (1986). Social and family factors in childhood hospital admission. Journal of Epidemiology & Community Health, 40(1), 50-58.

Barclay, L. M., Hutton, J. L., & Smith, J. Q. (2013). Refining a Bayesian network using a chain event graph. International Journal of Approximate Reasoning, 54(9), 1300-1309.

Standard variation of the covariance matrix

Description

Computation of an updated GBN object after a variation of the covariance matrix.

Usage

covariance_var(gbn, entry, delta)

Arguments

Details

Let the original Bayesian network have a Normal distribution \mathcal{N}(\mu,\Sigma) and let entry be equal to (i,j). For a variation of the covariance matrix by an amount \delta, a variation matrix D is constructed as

D_{k,l}=\left\{ \begin{array}{ll} \delta & \mbox{if } k=i, l=j\\ \delta & \mbox{if } l=i, k=j \\ 0 & \mbox{otherwise} \end{array} \right.

Then the resulting distribution after the variation is \mathcal{N}(\mu,\Sigma +D), assuming \Sigma+ D is positive semi-definite.

Value

If the resulting covariance is positive semi-definite, covariance_var returns an object of class GBN with an updated covariance matrix. Otherwise it returns an object of class npsd.gbn, which has the same components of GBN but also has a warning entry specifying that the covariance matrix is not positive semi-definite.

References

Gómez-Villegas, M. A., Maín, P., & Susi, R. (2007). Sensitivity analysis in Gaussian Bayesian networks using a divergence measure. Communications in Statistics—Theory and Methods, 36(3), 523-539.

Gómez-Villegas, M. A., Main, P., & Susi, R. (2013). The effect of block parameter perturbations in Gaussian Bayesian networks: Sensitivity and robustness. Information Sciences, 222, 439-458.

See Also

mean_var, model_pres_cov

Examples

covariance_var(synthetic_gbn,c(1,1),3)
covariance_var(synthetic_gbn,c(1,2),-0.4)

Co-variation schemes

Description

Functions that return an updated Bayesian network using the proportional, uniform and order-preserving co-variation schemes.

Usage

proportional_covar(bnfit, node, value_node, value_parents, new_value)

orderp_covar(bnfit, node, value_node, value_parents, new_value)

uniform_covar(bnfit, node, value_node, value_parents, new_value)

Arguments

Details

The Bayesian network on which parameter variation is being conducted should be expressed as a bn.fit object. The name of the node to be varied, its level and its parent's levels should be specified. The parameter variation specified by the function is:

P ( node = value_node | parents = value_parents ) = new_value

For orderp_covar, if two or more parameters in a distribution have the same value, the order is given by the one in the respective conditional probability table. Furthermore, the parameter associated to the largest probability of the conditional probability law cannot be varied.

Value

An object of class bn.fit with updated probabilities.

References

Laskey, K. B. (1995). Sensitivity analysis for probability assessments in Bayesian networks. IEEE Transactions on Systems, Man, and Cybernetics, 25(6), 901-909.

Renooij, S. (2014). Co-variation for sensitivity analysis in Bayesian networks: Properties, consequences and alternatives. International Journal of Approximate Reasoning, 55(4), 1022-1042.

Leonelli, M., & Riccomagno, E. (2022). A geometric characterization of sensitivity analysis in monomial models. International Journal of Approximate Reasoning, 151, 64-84. #'

Examples

proportional_covar(synthetic_bn, "y3", "2", c("2","1"), 0.3)
uniform_covar(synthetic_bn, "y2", "1", "2", 0.3)
orderp_covar(synthetic_bn, "y1", "1", NULL, 0.3)

Co-variation matrices

Description

Construction of model-preserving co-variation matrices for objects of class CI.

Usage

total_covar_matrix(ci, entry, delta)

col_covar_matrix(ci, entry, delta)

partial_covar_matrix(ci, entry, delta)

row_covar_matrix(ci, entry, delta)

Arguments

Details

Functions to compute total, partial, row-based and column-based co-variation matrices to ensure the conditional independences of the original Bayesian network hold after a variation. If no co-variation is required for model-preservation the functions return a matrix filled with ones (no co-variation).

Value

A co-variation matrix of the same size of the covariance matrix of CI.

References

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

See Also

model_pres_cov

Examples

total_covar_matrix(synthetic_ci,c(1,1),0.3)
total_covar_matrix(synthetic_ci,c(1,2),0.3)
partial_covar_matrix(synthetic_ci,c(1,2),0.3)
row_covar_matrix(synthetic_ci,c(1,2),0.3)
col_covar_matrix(synthetic_ci,c(1,2),0.3)

Pima Indian Diabetes Data

Description

Discretized version of the widely-used Pima Indians Diabetes Database

Format

A dataframe with 392 observations on the following 9 binary variables:

• PREG: number of times pregnant (low/high)

• GLUC: plasma glucose concentration (low/high)

• PRES: diastolic blood pressure (low/high)

• TRIC: triceps skin fold thickness (low/high)

• INS: 2-hour serum insulin (low/high)

• MASS: body mass index (low/high)

• PED: diabetes pedigree function (low/high)

• AGE: age (low/high)

• DIAB: test for diabetes (neg/pos)

Source

These data have been taken from the UCI Repository Of Machine Learning Databases. We chose this dataset because it best showcases the function of our monitors. However, we acknowledge that this data is used here without the consent of or compensation for the original Akimel O'odham participants.

Diameters in a Bayesian network

Description

Computation of the diameters of all conditional probability tables in a Bayesian network.

Usage

diameter(bnfit)

Arguments

Details

The diameter of a conditional probability table P with n rows p_1,\dots,p_n is

d^+(P)=\max_{i,j\leq n} d_V(p_i,p_j),

where d_V is the total variation distance between two probability mass functions over a sample space \mathcal{X}, i.e.

d_V(p_i,p_j)=\frac{1}{2}\sum_{x\in\mathcal{X}}|p_i(x)-p_j(x)|.

Value

A dataframe with the following columns: Nodes - the vertices of the BN; Diameter - the diameters of the associated conditional probability tables.

References

Leonelli, M., Smith, J. Q., & Wright, S. K. (2024). The diameter of a stochastic matrix: A new measure for sensitivity analysis in Bayesian networks. arXiv preprint arXiv:2407.04667.

Examples

diameter(travel)

Distance-weigthed influence

Description

Computation of the distance-weigthed influence in a Bayesian network

Usage

dwi(bn, node, w)

Arguments

Details

The distance-weigthed influence of a node X_j on an output node X_i in a Bayesian network is

DWI(X_j,X_i,w)= \sum_{s\in S_{ji}}w^{|s|},

where S_{ji} is the set of active trails between X_j and X_i, w\in(0,1] is an input parameter, and |s| is the length of the trail s.

Value

A dataframe with the following columns: Nodes - the vertices of the BN; Influence - the distance-weigthed influence of the corresponding node.

References

Albrecht, D., Nicholson, A. E., & Whittle, C. (2014). Structural sensitivity for the knowledge engineering of Bayesian networks. In Probabilistic Graphical Models (pp. 1-16). Springer International Publishing.

See Also

ewi, mutual_info

Examples

dwi(travel, "T", 0.5)

Strength of edges in a Bayesian network

Description

Computation of the measure of edge strength for all edges in a Bayesian networks

Usage

edge_strength(bnfit)

Arguments

Details

The measure of edge strength is defined as the largest diameter out of all conditional probability tables where all other parents but the considered one are fixed to a specific combination.

Value

A dataframe with first two columns the edge list of the bn.fit input object. The third column Edge.Strength reports the measure of edge strength.

References

Leonelli, M., Smith, J. Q., & Wright, S. K. (2024). The diameter of a stochastic matrix: A new measure for sensitivity analysis in Bayesian networks. arXiv preprint arXiv:2407.04667.

See Also

diameter

Examples

edge_strength(travel)

Edge-weigthed influence

Description

Computation of the edge-weigthed influence in a Bayesian network

Usage

ewi(bnfit, node)

Arguments

Details

The edge-weigthed influence of a node X_j on an output node X_i in a Bayesian network is

EWI(X_j,X_i)= \sum_{s\in S_{ji}}\left(\prod_{(k,l)\in s}\delta_{kl}\right)^{|s|},

where S_{ji} is the set of active trails between X_j and X_i, \delta_{kl} is the strength of an edge between X_k and X_l, and |s| is the length of the trail s.

Value

A dataframe with the following columns: Nodes - the vertices of the BN; Influence - the edge-weigthed influence of the corresponding node.

References

Leonelli, M., Smith, J. Q., & Wright, S. K. (2024). The diameter of a stochastic matrix: A new measure for sensitivity analysis in Bayesian networks. arXiv preprint arXiv:2407.04667.

See Also

mutual_info, dwi, edge_strength

Examples

ewi(travel, "T")

Final node monitors

Description

Marginal and conditional node monitors over the last observation of the data for all vertices of a Bayesian network using the full dataset

Usage

final_node_monitor(dag, df)

Arguments

Details

Consider a Bayesian network over variables Y_1,\dots,Y_m and suppose a dataset (\boldsymbol{y}_1,\dots,\boldsymbol{y}_n) has been observed, where \boldsymbol{y}_i=(y_{i1},\dots,y_{im}) and y_{ij} is the i-th observation of the j-th variable. Let p_n denote the marginal density of Y_j after the first n-1 observations have been processed. Define

E_n = \sum_{k=1}^Kp_n(d_k)\log(p_n(d_k)),

V_n = \sum_{k=1}^K p_n(d_k)\log^2(p_n(d_k))-E_n^2,

where (d_1,\dots,d_K) are the possible values of Y_j. The marginal node monitor for the vertex Y_j is defined as

Z_j=\frac{-\log(p_n(y_{ij}))- E_n}{\sqrt{V_n}}.

Higher values of Z_j can give an indication of a poor model fit for the vertex Y_j.

The conditional node monitor for the vertex Y_j is defined as

Z_j=\frac{-\log(p_n(y_{nj}|y_{n1},\dots,y_{n(j-1)},y_{n(j+1)},\dots,y_{nm}))- E_n}{\sqrt{V_n}},

where E_n and V_n are computed with respect to p_n(y_{nj}|y_{n1},\dots,y_{n(j-1)},y_{n(j+1)},\dots,y_{nm}). Again, higher values of Z_j can give an indication of a poor model fit for the vertex Y_j.

Value

A dataframe including the names of the vertices, the marginal node monitors and the conditional node monitors. It also return two plots where vertices with a darker color have a higher marginal z-score or conditional z-score, respectively, in absolute value.

References

Cowell, R. G., Dawid, P., Lauritzen, S. L., & Spiegelhalter, D. J. (2006). Probabilistic networks and expert systems: Exact computational methods for Bayesian networks. Springer Science & Business Media.

Cowell, R. G., Verrall, R. J., & Yoon, Y. K. (2007). Modeling operational risk with Bayesian networks. Journal of Risk and Insurance, 74(4), 795-827.

See Also

influential_obs, node_monitor, seq_node_monitor, seq_pa_ch_monitor

Examples

final_node_monitor(chds_bn, chds[1:100,])

Bayesian network on fire alarm system

Description

fire_alarm is a bn.fit object including a Bayesian network for a fire alarm system.

Usage

fire_alarm

Format

The Bayesian network fire_alarm includes the following nodes:

• Fire: two-level factor with levels TRUE and FALSE. It indicates presence or absence of a fire.

• Smoke: two level-factor with levels TRUE and FALSE. It indicates presence or absence of smoke.

• Alarm: three level-factor with levels TRUE, MALFUNCTION and FALSE. It indicates if the alarm is ringing, malfunctioning or not ringing.

• Tampering: two level-factor with levels TRUE and FALSE. It indicates if the alarm system has been tampered or not.

• Leaving: two level-factor with levels TRUE and FALSE. It indicates if the building is being evacuated or not.

• Report: two level-factor with levels TRUE and FALSE. It indicates if the incident has been reported or not.

Source

Hei Chan, Adnan Darwiche (2002). "When do numbers really matter?". Journal of Artificial Intelligence Research 17 (265-287).

Global monitor

Description

Negative marginal log-likelihood of the model

Usage

global_monitor(dag, df, alpha = "default")

Arguments

Value

A numerical value

References

Cowell, R. G., Dawid, P., Lauritzen, S. L., & Spiegelhalter, D. J. (2006). Probabilistic networks and expert systems: Exact computational methods for Bayesian networks. Springer Science & Business Media.

Cowell, R. G., Verrall, R. J., & Yoon, Y. K. (2007). Modeling operational risk with Bayesian networks. Journal of Risk and Insurance, 74(4), 795-827.

See Also

node_monitor, influential_obs, final_node_monitor, seq_node_monitor, seq_pa_ch_monitor

Examples

global_monitor(chds_bn, chds, 3)

Influential observations

Description

Influence of a single observation to the global monitor

Usage

influential_obs(dag, data, alpha = "default")

Arguments

Details

Consider a Bayesian network over variables Y_1,\dots,Y_m and suppose a dataset (\boldsymbol{y}_1,\dots,\boldsymbol{y}_n) has been observed, where \boldsymbol{y}_i=(y_{i1},\dots,y_{im}) and y_{ij} is the i-th observation of the j-th variable. Define \boldsymbol{y}_{-i}=(\boldsymbol{y}_1,\dots,\boldsymbol{y}_{i-1},\boldsymbol{y}_{i+1},\dots,\boldsymbol{y}_n). The influence of an observation to the global monitor is defined as

|\log(p(\boldsymbol{y}_1,\dots,\boldsymbol{y}_n)) - \log(p(\boldsymbol{y}_{-i}))|.

High values of this index denote observations that highly contribute to the likelihood of the model.

Value

A vector including the influence of each observation.

See Also

influential_obs, node_monitor, seq_node_monitor, seq_pa_ch_monitor

Examples

influential_obs(chds_bn, chds[1:100,], 3)

Math Marks Data

Description

Marks out of 100 for 88 students taking examinations in mechanics (C), vectors (C), algebra (0), analysis (O) and statistics (O), where C indicates closed and O indicates open book examination.

Usage

data(mathmarks)

Format

A dataframe with 88 observations on the following 5 variables

• mechanics: mark out of 100 for mechanics

• vectors: mark out of 100 for vectors

• algebra: mark out of 100 for algebra

• analysis: mark out of 100 for analysis

• statistics: mark out of 100 for statistics

Source

Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979) Multivariate Analysis. London: Academic Press.

Standard variation of the mean vector

Description

Computation of an updated GBN object after a variation of the mean vector.

Usage

mean_var(gbn, entry, delta)

Arguments

Details

Let the original Bayesian network have a Normal distribution \mathcal{N}(\mu,\Sigma) and let entry be equal to i. Let \mu_i be the i-th entry of \mu. For a variation of the mean by an amount \delta the resulting distribution is \mathcal{N}(\mu',\Sigma), where \mu' is equal to \mu except for the i-th entry which is equal to \mu+\delta.

Value

An object of class GBN with an updated mean vector.

References

Gómez-Villegas, M. A., Maín, P., & Susi, R. (2007). Sensitivity analysis in Gaussian Bayesian networks using a divergence measure. Communications in Statistics—Theory and Methods, 36(3), 523-539.

Gómez-Villegas, M. A., Main, P., & Susi, R. (2013). The effect of block parameter perturbations in Gaussian Bayesian networks: Sensitivity and robustness. Information Sciences, 222, 439-458.

See Also

covariance_var

Examples

mean_var(synthetic_gbn,2,3)

Model-Preserving co-variation

Description

Model-preserving co-variation for objects of class CI.

Usage

model_pres_cov(ci, type, entry, delta)

Arguments

Details

Let the original Bayesian network have a Normal distribution \mathcal{N}(\mu,\Sigma) and let entry be equal to (i,j). For a multiplicative variation of the covariance matrix by an amount \delta, a variation matrix \Delta is constructed as

\Delta_{k,l}=\left\{ \begin{array}{ll} \delta & \mbox{if } k=i, l=j\\ \delta & \mbox{if } l=i, k=j \\ 0 & \mbox{otherwise} \end{array} \right.

A co-variation matrix \tilde\Delta is then constructed and the resulting distribution after the variation is \mathcal{N}(\mu,\tilde\Delta\circ\Delta\circ\Sigma), assuming \tilde\Delta\circ\Delta\circ\Sigma is positive semi-definite and where \circ denotes the Schur (or element-wise) product. The matrix \tilde\Delta is so constructed to ensure that all conditional independence in the original Bayesian networks are retained after the parameter variation.

Value

If the resulting covariance is positive semi-definite, model_pres_cov returns an object of class CI with an updated covariance matrix. Otherwise it returns an object of class npsd.ci, which has the same components of CI but also has a warning entry specifying that the covariance matrix is not positive semi-definite.

References

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

See Also

covariance_var, covariation_matrix

Examples

model_pres_cov(synthetic_ci,"partial",c(1,3),1.1)
model_pres_cov(synthetic_ci,"partial",c(1,3),0.9)
model_pres_cov(synthetic_ci,"total",c(1,2),0.5)
model_pres_cov(synthetic_ci,"row",c(1,3),0.98)
model_pres_cov(synthetic_ci,"column",c(1,3),0.98)

Mutual information

Description

Computation of the mutual information in a Bayesian network

Usage

mutual_info(bnfit, node)

Arguments

Details

The mutual information between two variables X_j and X_i with sample spaces \mathcal{X}_i and \mathcal{X}_j, respectively, is equal to

\sum_{x_j\in\mathcal{X}_j}\sum_{x_i\in\mathcal{X}_i}p(x_i,x_j)\log\frac{p(x_i,x_j)}{p(x_i)p(x_j)}.

Value

A dataframe with the following columns: Nodes - the vertices of the BN; MutualInfo - the mutual information of the corresponding node.

References

Albrecht, D., Nicholson, A. E., & Whittle, C. (2014). Structural sensitivity for the knowledge engineering of Bayesian networks. In Probabilistic Graphical Models (pp. 1-16). Springer International Publishing.

See Also

ewi, dwi

Examples

mutual_info(travel, "T")

Node monitor

Description

Contribution of each vertex of a Bayesian network to the global monitor

Usage

node_monitor(dag, df, alpha = "default")

Arguments

Details

Consider a Bayesian network over variables Y_1,\dots,Y_m and suppose a dataset (\boldsymbol{y}_1,\dots,\boldsymbol{y}_n) has been observed, where \boldsymbol{y}_i=(y_{i1},\dots,y_{im}) and y_{ij} is the i-th observation of the j-th variable. The global monitor is defined as the negative log-likelihood of the model, i.e.

-\log(p(\boldsymbol{y}_1,\dots,\boldsymbol{y}_n))= - \sum_{j=1}^m\sum_{i=1}^n \log(p(y_{ij} | \pi_{ij})),

where \pi_{ij} is the value of the parents of Y_j for the i-th observation. The contribution of the j-th vertex to the global monitor is thus

-\sum_{i=1}^n\log(p(y_{ij}|\pi_{ij})).

Value

A dataframe including the name of the vertices and the contribution of the vertices to the global monitor. It also returns a plot where vertices with higher contributions in absolute value are darker.

References

Cowell, R. G., Dawid, P., Lauritzen, S. L., & Spiegelhalter, D. J. (2006). Probabilistic networks and expert systems: Exact computational methods for Bayesian networks. Springer Science & Business Media.

Cowell, R. G., Verrall, R. J., & Yoon, Y. K. (2007). Modeling operational risk with Bayesian networks. Journal of Risk and Insurance, 74(4), 795-827.

See Also

global_monitor, influential_obs, final_node_monitor, seq_node_monitor, seq_pa_ch_monitor

Examples

node_monitor(chds_bn, chds, 3)

Plotting methods

Description

Plotting methods for outputs of bnmonitor functions

Usage

## S3 method for class 'seq_marg_monitor'
plot(x, ...)

## S3 method for class 'CD'
plot(x, ...)

## S3 method for class 'seq_cond_monitor'
plot(x, ...)

## S3 method for class 'node_monitor'
plot(x, ...)

## S3 method for class 'influential_obs'
plot(x, ...)

## S3 method for class 'jeffreys'
plot(x, ...)

## S3 method for class 'kl'
plot(x, ...)

## S3 method for class 'final_node_monitor'
plot(x, which, ...)

## S3 method for class 'seq_pa_ch_monitor'
plot(x, ...)

## S3 method for class 'sensitivity'
plot(x, ...)

## S3 method for class 'fro'
plot(x, ...)

## S3 method for class 'diameter'
plot(x, ...)

## S3 method for class 'edgestrength'
plot(x, ...)

## S3 method for class 'mutualinfo'
plot(x, ...)

## S3 method for class 'dwi'
plot(x, ...)

## S3 method for class 'ewi'
plot(x, ...)

Arguments

Value

A plot specific to the object it is applied to.

Printing methods

Description

Printing methods for outputs of bnmonitor functions

Usage

## S3 method for class 'sensitivity'
print(x, ...)

## S3 method for class 'diameter'
print(x, ...)

## S3 method for class 'mutualinfo'
print(x, ...)

## S3 method for class 'dwi'
print(x, ...)

## S3 method for class 'ewi'
print(x, ...)

## S3 method for class 'kl'
print(x, ...)

## S3 method for class 'CD'
print(x, ...)

## S3 method for class 'fro'
print(x, ...)

## S3 method for class 'node_monitor'
print(x, ...)

## S3 method for class 'jeffreys'
print(x, ...)

## S3 method for class 'final_node_monitor'
print(x, ...)

## S3 method for class 'seq_cond_monitor'
print(x, ...)

## S3 method for class 'seq_pa_ch_monitor'
print(x, ...)

## S3 method for class 'seq_marg_monitor'
print(x, ...)

Arguments

Value

Printing specific to the object it is applied to.

Check for positive semi-definiteness after a perturbation

Description

psd_check returns a boolean to determine if the covariance matrix after a perturbation is positive semi-definite.

Usage

psd_check(x, ...)

## S3 method for class 'GBN'
psd_check(x, entry, delta, ...)

## S3 method for class 'CI'
psd_check(x, type, entry, delta, ...)

Arguments

Details

The details depend on the class the method psd_check is applied to.

Let \Sigma be the covariance matrix of a Gaussian Bayesian network and let D be a perturbation matrix acting additively. The perturbed covariance matrix \Sigma+D is positive semi-definite if

\rho(D)\leq \lambda_{\min}(\Sigma)

where \lambda_{\min} is the smallest eigenvalue end \rho is the spectral radius.

Value

A dataframe including the variations performed and the check for positive semi-definiteness.

Methods (by class)

• psd_check(GBN): psd_check for objects GBN

• psd_check(CI): psd_check for objects CI

References

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

Examples

psd_check(synthetic_gbn,c(2,4),-3)
psd_check(synthetic_gbn,c(2,3),seq(-1,1,0.1))
psd_check(synthetic_ci,"partial",c(2,4),0.95)
psd_check(synthetic_ci,"all",c(2,3),seq(0.9,1.1,0.01))

Sensitivity function

Description

sensitivity returns the sensitivity function for a probabilistic query of interest with respect to a parameter change defined by the user.

Usage

sensitivity(
  bnfit,
  interest_node,
  interest_node_value,
  evidence_nodes = NULL,
  evidence_states = NULL,
  node,
  value_node,
  value_parents,
  new_value,
  covariation = "proportional"
)

Arguments

Details

The Bayesian network on which parameter variation is being conducted should be expressed as a bn.fit object. The name of the node to be varied, its level and its parent's level should be specified. The parameter variation specified by the function is:

P ( node = value_node | parents = value_parents ) = new_value

and the probabilistic query of interest is:

P ( interest_node = interest_node_value | evidence_nodes = evidence_states )

Value

A dataframe with the varied parameter values and the output probabilities for the co-variation schemes selected. If plot = TRUE the function also returns a plot of the sensitivity function.

References

Coupé, V. M., & Van Der Gaag, L. C. (2002). Properties of sensitivity analysis of Bayesian belief networks. Annals of Mathematics and Artificial Intelligence, 36(4), 323-356.

Leonelli, M., Goergen, C., & Smith, J. Q. (2017). Sensitivity analysis in multilinear probabilistic models. Information Sciences, 411, 84-97.

See Also

covariation, sensquery

Examples

sensitivity(synthetic_bn, "y2", "3", node = "y1",value_node = "1",
 value_parents = NULL, new_value = "all", covariation = "all")
sensitivity(synthetic_bn, "y3", "1", "y2", "1", node = "y1", "1", NULL, 0.9, "all")

Sensitivity of probability query

Description

sensquery returns, for a given change in a probability of interest, the parameters' changes to achieve it together with the corresponding CD distances.

Usage

sensquery(
  bnfit,
  interest_node,
  interest_node_value,
  new_value,
  evidence_nodes = NULL,
  evidence_states = NULL
)

Arguments

Details

The Bayesian network should be expressed as a bn.fit object. The name of the node of the probability of interest, its level and the new value should be specified. Evidence could be also indicated. The probability of interest is specified as follows:

P ( interest_node = interest_node_value | evidence_nodes = evidence_states ) = new_value

Only the proportional co-variation scheme is used.

Value

A dataframe with the following columns: node - the vertex of the proposed change; Value node - the level of node to be changed; Value parents - the levels of the parent variables of node; Original value - the original probability defined by Node, Value node and Value parents; Suggested change - the new proposed value for the probability defined by Node, Value node and Value parents; CD distance - the CD distance between the original and new network with the Suggested change.

References

Chan, H., & Darwiche, A. (2002). When do numbers really matter?. Journal of Artificial Intelligence Research, 17, 265-287.

See Also

sensitivity

Examples

sensquery(synthetic_bn,"y3", "3", 0.3)

Sequential node monitors

Description

Sequential marginal and conditional node monitors for a vertex of a Bayesian network.

Usage

seq_marg_monitor(dag, df, node.name)

seq_cond_monitor(dag, df, node.name)

Arguments

Details

Consider a Bayesian network over variables Y_1,\dots,Y_m and suppose a dataset (\boldsymbol{y}_1,\dots,\boldsymbol{y}_n) has been observed, where \boldsymbol{y}_i=(y_{i1},\dots,y_{im}) and y_{ij} is the i-th observation of the j-th variable. Let p_i denote the marginal density of Y_j after the first i-1 observations have been processed. Define

E_i = \sum_{k=1}^Kp_i(d_k)\log(p_i(d_k)),

V_i = \sum_{k=1}^K p_i(d_k)\log^2(p_i(d_k))-E_i^2,

where (d_1,\dots,d_K) are the possible values of Y_j. The sequential marginal node monitor for the vertex Y_j is defined as

Z_{ij}=\frac{-\sum_{k=1}^i\log(p_k(y_{kj}))-\sum_{k=1}^i E_k}{\sqrt{\sum_{k=1}^iV_k}}.

Values of Z_{ij} such that |Z_{ij}|> 1.96 can give an indication of a poor model fit for the vertex Y_j after the first i-1 observations have been processed.

The sequential conditional node monitor for the vertex Y_j is defined as

Z_{ij}=\frac{-\sum_{k=1}^i\log(p_k(y_{kj}|y_{k1},\dots,y_{k(j-1)},y_{k(j+1)},\dots,y_{km}))-\sum_{k=1}^i E_k}{\sqrt{\sum_{k=1}^iV_k}},

where E_k and V_k are computed with respect to p_k(y_{kj}|y_{k1},\dots,y_{k(j-1)},y_{k(j+1)},\dots,y_{km}). Again, values of Z_{ij} such that |Z_{ij}|> 1.96 can give an indication of a poor model fit for the vertex Y_j.

Value

A vector including the scores Z_{ij}, either marginal or conditional.

References

Cowell, R. G., Dawid, P., Lauritzen, S. L., & Spiegelhalter, D. J. (2006). Probabilistic networks and expert systems: Exact computational methods for Bayesian networks. Springer Science & Business Media.

Cowell, R. G., Verrall, R. J., & Yoon, Y. K. (2007). Modeling operational risk with Bayesian networks. Journal of Risk and Insurance, 74(4), 795-827.

See Also

influential_obs, node_monitor, seq_node_monitor, seq_pa_ch_monitor

Examples

seq_marg_monitor(chds_bn, chds[1:100,], "Events")
seq_marg_monitor(chds_bn, chds[1:100,], "Admission")

Sequential parent-child node monitors

Description

Sequential node monitor for a vertex of a Bayesian network for a specific configuration of its parents

Usage

seq_pa_ch_monitor(dag, df, node.name, pa.names, pa.val, alpha = "default")

Arguments

Details

Consider a Bayesian network over variables Y_1,\dots,Y_m and suppose a dataset (\boldsymbol{y}_1,\dots,\boldsymbol{y}_n) has been observed, where \boldsymbol{y}_i=(y_{i1},\dots,y_{im}) and y_{ij} is the i-th observation of the j-th variable. Consider a configuration \pi_j of the parents and consider the sub-vector \boldsymbol{y}'=(\boldsymbol{y}_1',\dots,\boldsymbol{y}_{N'}') of (\boldsymbol{y}_1,\dots,\boldsymbol{y}_n) including observations where the parents of Y_j take value \pi_j only. Let p_i(\cdot|\pi_j) be the conditional distribution of Y_j given that its parents take value \pi_j after the first i-1 observations have been processed. Define

E_i = \sum_{k=1}^Kp_i(d_k|\pi_j)\log(p_i(d_k|\pi_j)),

V_i = \sum_{k=1}^K p_i(d_k|\pi_j)\log^2(p_i(d_k|\pi_j))-E_i^2,

where (d_1,\dots,d_K) are the possible values of Y_j. The sequential parent-child node monitor for the vertex Y_j and parent configuration \pi_j is defined as

Z_{ij}=\frac{-\sum_{k=1}^i\log(p_k(y_{kj}'|\pi_j))-\sum_{k=1}^i E_k}{\sqrt{\sum_{k=1}^iV_k}}.

Values of Z_{ij} such that |Z_{ij}|> 1.96 can give an indication of a poor model fit for the vertex Y_j after the first i-1 observations have been processed.

Value

A vector including the scores Z_{ij}.

References

Cowell, R. G., Dawid, P., Lauritzen, S. L., & Spiegelhalter, D. J. (2006). Probabilistic networks and expert systems: Exact computational methods for Bayesian networks. Springer Science & Business Media.

Cowell, R. G., Verrall, R. J., & Yoon, Y. K. (2007). Modeling operational risk with Bayesian networks. Journal of Risk and Insurance, 74(4), 795-827.

See Also

influential_obs, node_monitor, seq_node_monitor, seq_pa_ch_monitor

Examples

seq_pa_ch_monitor(chds_bn, chds, "Events", "Social", "High", 3)

A synthetic Bayesian network

Description

synthetic_bn is a bn.fit object for a simple Bayesian network involving three variables.

Usage

synthetic_bn

Format

The Bayesian network bnsens_example comprehends the following nodes:

• y1: three-level factor with levels 1, 2, 3.

• y2: three-level factor with levels 1, 2, 3.

• y3: three-level factor with levels 1, 2, 3.

Source

Leonelli, M., & Riccomagno, E. (2022). A geometric characterization of sensitivity analysis in monomial models. International Journal of Approximate Reasoning, 151, 64-84.

A synthetic continuous Bayesian network

Description

A synthetic continuous Bayesian network

Usage

synthetic_gbn

synthetic_ci

Format

A continuous Bayesian networks over four variables ("y1", "y2", "y3", "y4"), embedding the statement "y1" independent of "y3" given "y2". The Bayesian network is available both as an object of class GBN and as an object of class CI.

An object of class GBN of length 3.

An object of class CI of length 4.

Source

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

Bayesian network on travel survey

Description

travel is a bn.fit object for the Bayesian network on a traveling preferences survey.

Usage

travel

Format

The Bayesian network travel includes the following nodes:

• A: three-level factor with levels young, adult, old. It indicates the age of an individual.

• S: two level-factor with levels M (male) and F (female). It indicates the gender of an individual.

• E: two level-factor with levels high and uni. It indicates the education level of an individual.

• O: two level-factor with levels emp (employed) and self (self-employed). It indicates the occupation of an individual.

• R: two level-factor with levels small and big. It indicates the size of the residence of an individual.

• T: three level-factor with levels car, train and other. It indicates the preferred mean of transportation by an individual.

Source

Scutari, M., & Denis, J. B. (2014). Bayesian networks: with examples in R. Chapman and Hall/CRC.