Bayesian network structure learning.

Description

Bayesian network learning via constraint-based and score-based algorithms.

Details

Package:	bnlearn
Type:	Package
Version:	1.3
Date:	2009-03-28
License:	GPLv2 or later

This package implements some constraint-based algorithms for learning the structure of Bayesian networks. Also known as conditional independence learners, they are all optimized derivatives of the Inductive Causation algorithm (Verma and Pearl, 1991).

These algorithms differ in the way they detect the Markov blankets of the variables, which in turn are used to compute the structure of the Bayesian network. Proofs of correctness are present in the respective papers.

A score-based learning algorithm (greedy search via hill-climbing) is implemented as well for comparison purposes.

Available constraint-based learning algorithms

• Grow-Shrink (gs): based on the Grow-Shrink Markov Blanket algorithm, a forward-selection technique for Markov blanket detection (Margaritis, 2003).
• Incremental Association (iamb): based on the Markov blanket detection algorithm of the same name, which uses stepwise selection (Tsamardinos et al., 2003).
• Fast Incremental Association (fast.iamb): a variant of IAMB which uses speculative forward-selection to reduce the number of conditional independence tests (Yaramakala and Margaritis, 2005).
• Interleaved Incremental Association (inter.iamb): another variant of IAMB which interleaves the backward selection with the forward one to avoid false positives (i.e. nodes erroneously included in the Markov blanket) (Yaramakala and Margaritis, 2005).
• 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 current conditioning variables (Tsamardinos, Brown and Aliferis, 2006). It learns the underlying structure of the Bayesian networks (all the arcs are undirected, no attempt is made to detect their orientation).

This package includes three implementations of each algorithm:

• an optimized implementation (used when the optimized parameter is set to TRUE), which uses backtracking to roughly halve the number of independence tests.
• 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 snow package. See snow integration for a sample usage.

The computational complexity of these algorithms is polynomial in the number of tests, usually O(N^2) (O(N^4) in the worst case scenario). The 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. Random restart with a configurable number of perturbing operations is implemented.

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:

• categorical data (multinomial distribution)
  • mutual information (mi): 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.
  • Monte Carlo mutual information (mc-mi): a Monte Carlo implementation of the mutual information, which does not rely on the asymptotic chi-square distribution.
  • fast mutual information (fmi): a variant of the mutual information which is set to zero when there aren't at least five data per parameter.
  • Pearson's chi-square (x2): the classical Pearson's Chi-squared test for contingency tables.
  • Monte Carlo Pearson's chi-square (mc-x2): a Monte Carlo implementation of the chi-square test, which does not rely on the asymptotic chi-square distribution.
  • Akaike Information Criterion (aict): an experimental AIC-based independence test, computed comparing the mutual information and the expected information gain.

• numeric data (multivariate normal distribution)
  • linear correlation (cor): linear correlation.
  • Monte Carlo linear correlation (mc-cor): a Monte Carlo implementation of the linear correlation test, which does not rely on the asymptotic Student's t distribution.
  • Fisher's Z (zf): 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).
  • Monte Carlo Fisher's Z (mc-zf): a Monte Carlo implementation of Fisher's Z test, which does not rely on the asymptotic normal distribution.
  • mutual information (mi-g): an information-theoretic distance measure. Again it's proportional to the log-likelihood ratio (they differ by a 2n factor).
  • Monte Carlo mutual information (mc-mi-g): a Monte Carlo implementation of the mutual information for Gaussian data, which does not rely on the asymptotic chi-square distribution.

Available network scores

Available scores (and the respective labels) are:

• categorical data (multinomial distribution)
  • the multinomial likelihood (lik) score.
  • the multinomial loglikelihood (loglik) score, which is equivalent to the entropy measure.
  • the Akaike Information Criterion score (aic).
  • the Bayesian Information Criterion score (bic), which is equivalent to the Minimum Description Length (MDL).
  • the logarithm of the Bayesian Dirichlet equivalent score (bde o dir), a score equivalent Dirichlet posterior density.
  • the logarithm of the K2 score (k2), a Dirichlet posterior density (not score equivalent).

• numeric data (multivariate normal distribution)
  • a score equivalent {Gaussian posterior density} (bge).

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.

Error detection and correction: the strict mode

Optimized implementations of the constraint-based algorithms rely heavily on backtracking to reduce the number of tests needed by the learning procedure. This approach may hide errors either in the Markov blanket or the neighbourhood detection phase in some particular cases, 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 the Markov blanket and neighbour detection 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, which may help to retrieve a good model even 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).

Author(s)

Marco Scutari
Department of Statistical Sciences
University of Padova

Maintainer: Marco Scutari marco.scutari@gmail.com

References

A. Agresti. Categorical Data Analysis. John Wiley & Sons, Inc., 2002.

K. Korb and A. Nicholson. Bayesian artificial intelligence. Chapman and Hall, 2004.

D. Margaritis. Learning Bayesian Network Model Structure from Data. PhD thesis, School of Computer Science, Carnegie-Mellon University, Pittsburgh, PA, May 2003. Available as Technical Report CMU-CS-03-153.

J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann Publishers Inc., 1988.

I. Tsamardinos, C. F. Aliferis, and A. Statnikov. Algorithms for large scale Markov blanket discovery. In Proceedings of the Sixteenth International Florida Artificial Intelligence Research Society Conference, pages 376-381. AAAI Press, 2003.

I. Tsamardinos, C. F. Aliferis, A. Statnikov. Time and sample efficient discovery of Markov blankets and direct causal relations. Proceedings of the Ninth International Conference on Knowledge Discovery and Data Mining KDD, pages 673-8, 2003.

I. Tsamardinos, L. E. Brown, C. Aliferis. The max-min hill-climbing Bayesian network learning algorithm. Machine Learning, 65(1), pages 31-78. Kluwer Academic Publishers, 2006.

S. Yaramakala, D. Margaritis. Speculative Markov Blanket Discovery for Optimal Feature Selection. In Proceedings of the Fifth IEEE International Conference on Data Mining, pages 809-812. IEEE Computer Society, 2005.

Examples

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?
compare(res, res2)
# [1] TRUE
# how many tests each of the two algorithms used?
res$learning$ntests
# [1] 41
res2$learning$ntests
# [1] 50
# and the unoptimized implementation of these algorithms?
## Not run: gs(learning.test, optimized = FALSE)$learning$ntests
# [1] 90
## Not run: iamb(learning.test, optimized = FALSE)$learning$ntests
# [1] 116

## 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
#   from to
# 1    B  F
# 2    F  B
res3 = gs(learning.test, blacklist = blacklist)
plot(res3)
# force E - F direction (E -> F).
whitelist = data.frame(from = c("E"), to = c("F"))
whitelist
#   from to
# 1    E  F
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()
## End(Not run)
# if something seems wrong, try the unoptimized version
# in strict mode (inconsistencies trigger errors):
## Not run: 
res = gs(learning.test, optimized = FALSE, strict = TRUE, debug = TRUE)
## End(Not run)
# or disable strict mode to let the algorithm fix errors on the fly:
## Not run: 
res = gs(learning.test, optimized = FALSE, strict = FALSE, debug = TRUE)
## End(Not run)