Measure the strength of the probabilistic relationships expressed by the arcs of a Bayesian network, and use model averaging to build a network containing only the significant arcs.
# strength of the arcs present in x.
arc.strength(x, data, criterion = NULL, ..., debug = FALSE)
# strength of all possible arcs, as learned from bootstrapped data.
boot.strength(data, cluster, R = 200, m = nrow(data),
algorithm, algorithm.args = list(), cpdag = TRUE, debug = FALSE)
# strength of all possible arcs, from a list of custom networks.
custom.strength(networks, nodes, weights = NULL, cpdag = TRUE, debug = FALSE)
# strength of all possible arcs, computed using Bayes factors.
bf.strength(x, data, score, ..., debug = FALSE)# average arc strengths.
# S3 method for bn.strength
mean(x, ..., weights = NULL)
# averaged network structure.
averaged.network(strength, threshold)
# strength threshold for inclusion in the averaged network structure.
inclusion.threshold(strength)
arc.strength(), boot.strength(), custom.strength(),
bf.strength() and mean() return an object of class
bn.strength; boot.strength() and custom.strength() also
include information about the relative probabilities of arc directions.
averaged.network() returns an object of class bn.
See bn.strength class and bn-class for details.
an object of class bn.strength (for mean()) or of
class bn (for all other functions).
a list, containing either object of class bn or arc
sets (matrices or data frames with two columns, optionally labeled "from"
and "to"); or an object of class bn.kcv or bn.kcv.list
from bn.cv().
a data frame containing the data the Bayesian network was
learned from (for arc.strength()) or that will be used to compute
the arc strengths (for boot.strength() and bf.strength()).
an optional cluster object from package parallel.
an object of class bn.strength, see below.
a numeric value, the minimum strength required for an
arc to be included in the averaged network. The default value is the
threshold attribute of the strength argument.
a vector of character strings, the labels of the nodes in the network.
a character string. For arc.strength(), the
label of a score function or an independence test; see
network scores for details.
For bf.strength(), the
label of the score used to compute the Bayes factors; see BF
for details.
a positive integer, the number of bootstrap replicates.
a positive integer, the size of each bootstrap replicate.
a vector of non-negative numbers, to be used as weights
when averaging arc strengths (in mean()) or network structures (in
custom.strength()) to compute strength coefficients. If NULL,
weights are assumed to be uniform.
a boolean value. If TRUE the (PDAG of) the equivalence
class is used instead of the network structure itself. It should make it
easier to identify score-equivalent arcs.
a character string, the structure learning algorithm to be
applied to the bootstrap replicates. See structure learning
and the documentation of each algorithm for details.
a list of extra arguments to be passed to the learning algorithm.
in arc.strength(), the additional tuning parameters for
the network score (if criterion is the label of a score function,
see score for details), the conditional independence test
(currently the only one is B, the number of permutations). In
mean(), additional objects of class bn.strength to average.
a boolean value. If TRUE a lot of debugging output is
printed; otherwise the function is completely silent.
Marco Scutari
arc.strength() computes a measure of confidence or strength for each
arc, while keeping fixed the rest of the network structure.
If criterion is a conditional independence test, the strength is a
p-value (so the lower the value, the stronger the relationship). The
conditional independence test would be that to drop the arc from the network.
The only two possible additional arguments are alpha, which sets the
significance threshold that is used in strength.plot(); and B,
the number of permutations to be generated for each permutation test.
If criterion is the label of a score function, the strength is
measured by the score gain/loss which would be caused by the arc's removal.
In other words, it is the difference between the score of the network
in which the arc is not present and the score of the network in which the arc
is present. Negative values correspond to decreases in the network score
and positive values correspond to increases in the network score (the stronger
the relationship, the more negative the difference). There may be additional
arguments depending on the choice of the score, see score for
details. The significance threshold is set to 0.
boot.strength() estimates the strength of each arc as its empirical
frequency over a set of networks learned from bootstrap samples. It computes
the probability of each arc (modulo its direction) and the probabilities of
each arc's directions conditional on the arc being present in the graph (in
either direction). The significance threshold is computed automatically from
the strength estimates.
bf.strength() estimates the strength of each arc using Bayes factors
to overcome the fact that Bayesian posterior scores are not normalised, and
uses the latter to estimate the probabilities of all possible states of an
arc given the rest of the network. The significance threshold is set to
1.
custom.strength() takes a list of networks and estimates arc strength
in the same way as
boot.strength().
Model averaging is supported for objects of class bn.strength returned
by boot.strength, custom.strength and
bf.strength. The returned network contains the arcs whose
strength is greater than the threshold attribute of the
bn.strength object passed to averaged.network().
for model averaging and boostrap strength (confidence):
Friedman N, Goldszmidt M, Wyner A (1999). "Data Analysis with Bayesian Networks: A Bootstrap Approach". Proceedings of the 15th Annual Conference on Uncertainty in Artificial Intelligence, 196--201.
for the computation of the bootstrap strength (confidence) significance threshold:
Scutari M, Nagarajan R (2013). "On Identifying Significant Edges in Graphical Models of Molecular Networks". Artificial Intelligence in Medicine, 57(3):207--217.
strength.plot, choose.direction,
score, ci.test.
data(learning.test)
res = hc(learning.test)
arc.strength(res, learning.test)
if (FALSE) {
arcs = boot.strength(learning.test, algorithm = "hc")
arcs[(arcs$strength > 0.85) & (arcs$direction >= 0.5), ]
averaged.network(arcs)
start = random.graph(nodes = names(learning.test), num = 50)
netlist = lapply(start, function(net) {
hc(learning.test, score = "bde", iss = 10, start = net) })
arcs = custom.strength(netlist, nodes = names(learning.test),
cpdag = FALSE)
arcs[(arcs$strength > 0.85) & (arcs$direction >= 0.5), ]
modelstring(averaged.network(arcs))
}
bf.strength(res, learning.test, score = "bds", prior = "marginal")
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