Bayesian network structure learning, parameter learning and inference

Description

Bayesian network structure learning (via constraint-based, score-based and hybrid algorithms), parameter learning (via ML and Bayesian estimators) and inference.

Details

Package: bnlearn
Type: Package
Version: 4.0
Date: 2016-05-12
License: GPLv2 or later

This package implements some algorithms for learning the structure of Bayesian networks.

Constraint-based algorithms, also known as conditional independence learners, are all optimized derivatives of the Inductive Causation algorithm (Verma and Pearl, 1991). These algorithms use conditional independence tests to detect the Markov blankets of the variables, which in turn are used to compute the structure of the Bayesian network.

Score-based learning algorithms are general purpose heuristic optimization algorithms which rank network structures with respect to a goodness-of-fit score.

Hybrid algorithms combine aspects of both constraint-based and score-based algorithms, as they use conditional independence tests (usually to reduce the search space) and network scores (to find the optimal network in the reduced space) at the same time.

Several functions for parameter estimation, parametric inference, bootstrap, cross-validation and stochastic simulation are available. Furthermore, advanced plotting capabilities are implemented on top of the Rgraphviz and lattice packages.

Available constraint-based learning algorithms

  • Grow-Shrink (gs): based on the Grow-Shrink Markov Blanket, the first (and simplest) Markov blanket detection algorithm used in a structure learning algorithm.

  • Incremental Association (iamb): based on the Markov blanket detection algorithm of the same name, which is based on a two-phase selection scheme (a forward selection followed by an attempt to remove false positives).

  • Fast Incremental Association (fast.iamb): a variant of IAMB which uses speculative stepwise forward selection to reduce the number of conditional independence tests.

  • Interleaved Incremental Association (inter.iamb): another variant of IAMB which uses forward stepwise selection to avoid false positives in the Markov blanket detection phase.

This package includes three implementations of each algorithm:

  • an optimized implementation (used when the optimized parameter is set to TRUE), which uses backtracking to initialize the learning process of each node.

  • an unoptimized implementation (used when the optimized parameter is set to FALSE) which is better at uncovering possible erratic behaviour of the statistical tests.

  • a cluster-aware implementation, which requires a running cluster set up with the makeCluster function from the parallel package. See parallel integration for a sample usage.

The computational complexity of these algorithms is polynomial in the number of tests, usually O(N^2) (but super-exponential in the worst case scenario), where N is the number of variables. Execution time scales linearly with the size of the data set.

Available score-based learning algorithms

  • Hill-Climbing (hc): a hill climbing greedy search on the space of the directed graphs. The optimized implementation uses score caching, score decomposability and score equivalence to reduce the number of duplicated tests.

  • Tabu Search (tabu): a modified hill-climbing able to escape local optima by selecting a network that minimally decreases the score function.

Random restart with a configurable number of perturbing operations is implemented for both algorithms.

Available hybrid learning algorithms

  • Max-Min Hill-Climbing (mmhc): a hybrid algorithm which combines the Max-Min Parents and Children algorithm (to restrict the search space) and the Hill-Climbing algorithm (to find the optimal network structure in the restricted space).

  • Restricted Maximization (rsmax2): a more general implementation of the Max-Min Hill-Climbing, which can use any combination of constraint-based and score-based algorithms.

Other (constraint-based) local discovery algorithms

These algorithms learn the structure of the undirected graph underlying the Bayesian network, which is known as the skeleton of the network or the (partial) correlation graph. Therefore all the arcs are undirected, and no attempt is made to detect their orientation. They are often used in hybrid learning algorithms.

  • Max-Min Parents and Children (mmpc): a forward selection technique for neighbourhood detection based on the maximization of the minimum association measure observed with any subset of the nodes selected in the previous iterations.

  • Hiton Parents and Children (si.hiton.pc): a fast forward selection technique for neighbourhood detection designed to exclude nodes early based on the marginal association. The implementation follows the Semi-Interleaved variant of the algorithm.

  • Chow-Liu (chow.liu): an application of the minimum-weight spanning tree and the information inequality. It learns the tree structure closest to the true one in the probability space.

  • ARACNE (aracne): an improved version of the Chow-Liu algorithm that is able to learn polytrees.

All these algorithms have three implementations (unoptimized, optimized and cluster-aware) like other constraint-based algorithms.

Bayesian Network classifiers

The algorithms are aimed at classification, and favour predictive power over the ability to recover the correct network structure. The implementation in bnlearn assumes that all variables, including the classifiers, are discrete.

  • Naive Bayes (naive.bayes): a very simple algorithm assuming that all classifiers are independent and using the posterior probability of the target variable for classification.

  • Tree-Augmented Naive Bayes (tree.bayes): an improvement over naive Bayes, this algorithms uses Chow-Liu to approximate the dependence structure of the classifiers.

Available (conditional) independence tests

The conditional independence tests used in constraint-based algorithms in practice are statistical tests on the data set. Available tests (and the respective labels) are:

  • discrete case (categorical variables)

    • mutual information: an information-theoretic distance measure. It's proportional to the log-likelihood ratio (they differ by a 2n factor) and is related to the deviance of the tested models. The asymptotic chi-square test (mi and mi-adf, with adjusted degrees of freedom), the Monte Carlo permutation test (mc-mi), the sequential Monte Carlo permutation test (smc-mi), and the semiparametric test (sp-mi) are implemented.

    • shrinkage estimator for the mutual information (mi-sh): an improved asymptotic chi-square test based on the James-Stein estimator for the mutual information.

    • Pearson's X^2: the classical Pearson's X^2 test for contingency tables. The asymptotic chi-square test (x2 and x2-adf, with adjusted degrees of freedom), the Monte Carlo permutation test (mc-x2), the sequential Monte Carlo permutation test (smc-x2) and semiparametric test (sp-x2) are implemented.

  • discrete case (ordered factors)

    • Jonckheere-Terpstra: a trend test for ordinal variables. The asymptotic normal test (jt), the Monte Carlo permutation test (mc-jt) and the sequential Monte Carlo permutation test (smc-jt) are implemented.

  • continuous case (normal variables)

    • linear correlation: Pearson's linear correlation. The exact Student's t test (cor), the Monte Carlo permutation test (mc-cor) and the sequential Monte Carlo permutation test (smc-cor) are implemented.

    • Fisher's Z: a transformation of the linear correlation with asymptotic normal distribution. Used by commercial software (such as TETRAD II) for the PC algorithm (an R implementation is present in the pcalg package on CRAN). The asymptotic normal test (zf), the Monte Carlo permutation test (mc-zf) and the sequential Monte Carlo permutation test (smc-zf) are implemented.

    • mutual information: an information-theoretic distance measure. Again it is proportional to the log-likelihood ratio (they differ by a 2n factor). The asymptotic chi-square test (mi-g), the Monte Carlo permutation test (mc-mi-g) and the sequential Monte Carlo permutation test (smc-mi-g) are implemented.

    • shrinkage estimator for the mutual information (mi-g-sh): an improved asymptotic chi-square test based on the James-Stein estimator for the mutual information.

  • hybrid case (mixed discrete and normal variables)

    • mutual information: an information-theoretic distance measure. Again it is proportional to the log-likelihood ratio (they differ by a 2n factor). Only the asymptotic chi-square test (mi-cg) is implemented.

Available network scores

Available scores (and the respective labels) are:

  • discrete case (categorical variables)

    • the multinomial log-likelihood (loglik) score, which is equivalent to the entropy measure used in Weka.

    • the Akaike Information Criterion score (aic).

    • the Bayesian Information Criterion score (bic), which is equivalent to the Minimum Description Length (MDL) and is also known as Schwarz Information Criterion.

    • the logarithm of the Bayesian Dirichlet equivalent score (bde), a score equivalent Dirichlet posterior density.

    • the logarithm of the modified Bayesian Dirichlet equivalent score (mbde) for mixtures of experimental and observational data (not score equivalent).

    • the logarithm of the K2 score (k2), a Dirichlet posterior density (not score equivalent).

  • continuous case (normal variables)

    • the multivariate Gaussian log-likelihood (loglik-g) score.

    • the corresponding Akaike Information Criterion score (aic-g).

    • the corresponding Bayesian Information Criterion score (bic-g).

    • a score equivalent Gaussian posterior density (bge).

  • hybrid case (mixed discrete and normal variables)

    • the conditional linear Gaussian log-likelihood (loglik-cg) score.

    • the corresponding Akaike Information Criterion score (aic-cg).

    • the corresponding Bayesian Information Criterion score (bic-cg).

Whitelist and blacklist support

All learning algorithms support arc whitelisting and blacklisting:

  • blacklisted arcs are never present in the graph.

  • arcs whitelisted in one direction only (i.e. A -> B is whitelisted but B -> A is not) have the respective reverse arcs blacklisted, and are always present in the graph.

  • arcs whitelisted in both directions (i.e. both A -> B and B -> A are whitelisted) are present in the graph, but their direction is set by the learning algorithm.

Any arc whitelisted and blacklisted at the same time is assumed to be whitelisted, and is thus removed from the blacklist.

In algorithms that learn undirected graphs, such as ARACNE and Chow-Liu, an arc must be blacklisted in both directions to blacklist the underlying undirected arc.

Error detection and correction: the strict mode

Optimized implementations of constraint-based algorithms rely heavily on backtracking to reduce the number of tests needed by the learning algorithm. This approach may sometimes hide errors either in the Markov blanket or the neighbourhood detection steps, such as when hidden variables are present or there are external (logical) constraints on the interactions between the variables.

On the other hand, in the unoptimized implementations of constraint-based algorithms the learning of the Markov blanket and neighbourhood of each node is completely independent from the rest of the learning process. Thus it may happen that the Markov blanket or the neighbourhoods are not symmetric (i.e. A is in the Markov blanket of B but not vice versa), or that some arc directions conflict with each other.

The strict parameter enables some measure of error correction for such inconsistencies, which may help to retrieve a good model when the learning process would otherwise fail:

  • if strict is set to TRUE, every error stops the learning process and results in an error message.

  • if strict is set to FALSE:

    1. v-structures are applied to the network structure in lowest-p-value order; if any arc is already oriented in the opposite direction, the v-structure is discarded.

    2. nodes which cause asymmetries in any Markov blanket are removed from that Markov blanket; they are treated as false positives.

    3. nodes which cause asymmetries in any neighbourhood are removed from that neighbourhood; again they are treated as false positives (see Tsamardinos, Brown and Aliferis, 2006).

    Each correction results in a warning.

Author(s)

Marco Scutari
UCL Genetics Institute (UGI)
University College London

Maintainer: Marco Scutari marco.scutari@gmail.com

References

(a BibTeX file with all the references cited throughout this manual is present in the ‘bibtex’ directory of this package)

Nagarajan R, Scutari M, Lebre S (2013). "Bayesian Networks in R with Applications in Systems Biology". Springer.

Scutari M (2010). "Learning Bayesian Networks with the bnlearn R Package". Journal of Statistical Software, 35(3), 1-22. URL http://www.jstatsoft.org/v35/i03/.

Koller D, Friedman N (2009). Probabilistic Graphical Models: Principles and Techniques. MIT Press.

Korb K, Nicholson AE (2010). Bayesian Artificial Intelligence. Chapman & Hall/CRC, 2nd edition.

Pearl J (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann.

Examples

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library(bnlearn)
data(learning.test)

## Simple learning
# first try the Grow-Shrink algorithm
res = gs(learning.test)
# plot the network structure.
plot(res)
# now try the Incremental Association algorithm.
res2 = iamb(learning.test)
# plot the new network structure.
plot(res2)
# the network structures seem to be identical, don't they?
all.equal(res, res2)
# how many tests each of the two algorithms used?
ntests(res)
ntests(res2)
# and the unoptimized implementation of these algorithms?
## Not run: ntests(gs(learning.test, optimized = FALSE))
## Not run: ntests(iamb(learning.test, optimized = FALSE))

## Greedy search
res = hc(learning.test)
plot(res)

## Another simple example (Gaussian data)
data(gaussian.test)
# first try the Grow-Shrink algorithm
res = gs(gaussian.test)
plot(res)

## Blacklist and whitelist use
# the arc B - F should not be there?
blacklist = data.frame(from = c("B", "F"), to = c("F", "B"))
blacklist
res3 = gs(learning.test, blacklist = blacklist)
plot(res3)
# force E - F direction (E -> F).
whitelist = data.frame(from = c("E"), to = c("F"))
whitelist
res4 = gs(learning.test, whitelist = whitelist)
plot(res4)
# use both blacklist and whitelist.
res5 = gs(learning.test, whitelist = whitelist, blacklist = blacklist)
plot(res5)

## Debugging
# use the debugging mode to see the learning algorithms
# in action.
res = gs(learning.test, debug = TRUE)
res = hc(learning.test, debug = TRUE)
# log the learning process for future reference.
## Not run: 
sink(file = "learning-log.txt")
res = gs(learning.test, debug = TRUE)
sink()
# if something seems wrong, try the unoptimized version
# in strict mode (inconsistencies trigger errors):
res = gs(learning.test, optimized = FALSE, strict = TRUE, debug = TRUE)
# or disable strict mode to let the algorithm fix errors on the fly:
res = gs(learning.test, optimized = FALSE, strict = FALSE, debug = TRUE)

## End(Not run)

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