pcalg (version 2.0-3)

gds: Greedy DAG Search to Estimate Markov Equivalence Class of DAG

Description

Estimate the observational or interventional essential graph representing the Markov equivalence class of a DAG by greedily optimizing a score function in the space of DAGs. In practice, greedy search should always be done in the space of equivalence classes instead of DAGs, giving the functions gies or ges the preference over gds.

Usage

gds(p, targets, score, verbose = FALSE, ...)

Arguments

p

number of variables.

targets

a list of intervention targets (cf. details). A list of vectors, each vector listing the vertices of one intervention target.

score

an instance of a class derived from Score.

verbose

if TRUE, detailed output is provided.

...

additional arguments for debugging purposes and fine tuning.

Value

• gds returns a list with the following two components:
• essgraph
  An object of class EssGraph containing an estimate of the equivalence class of the underlying DAG.
• repr
  An object of a class derived from ParDAG containing a (random) representative of the estimated equivalence class.

encoding

UTF-8

concept

• greedy DAG search
• essential graph

Details

This function estimates the observational or interventional Markov equivalence class of a DAG based on a data sample with interventional data originating from various interventions and possibly observational data. The intervention targets used for data generation must be specified by the argument targets as a list of (integer) vectors listing the intervened vertices; observational data is specified by an empty set, i.e. a vector of the form integer(0). As an example, if data contains observational samples as well as samples originating from an intervention at vertices 1 and 4, the intervention targets must be specified as list(integer(0), as.integer(1), as.integer(c(1, 4))). An interventional Markov equivalence class of DAGs can be uniquely represented by a partially directed graph called interventional essential graph. Its edges have the following interpretation:

1. a directed edge$a \longrightarrow b$stands for an arrow that has the same orientation in all representatives of the interventional Markov equivalence class;
2. an undirected edge a -- b stands for an arrow that is oriented in one way in some representatives of the equivalence class and in the other way in other representatives of the equivalence class.

Note that when plotting the object, undirected and bidirected edges are equivalent. Greedy DAG search (GDS) maximizes a score function (typically the BIC, passed to the function via the argument score) of a DAG in three phases, starting from the empty DAG: [object Object],[object Object],[object Object] GIES cycles through these three phases until no augmentation of the score is possible any more. In the end, gds returns the (interventional or observational) essential graph of the last visited DAG. It is well-known that a greedy search in the space of DAGs instead of essential graphs is more prone to be stuck in local optima of the score function and hence expected to yield worse estimation results than GIES (function gies) or GES (function ges) (Chickering, 2002; Hauser and Bühlmann, 2012). The function gds is therefore not of practical use, but can be used to compare causal inference algorithms to an elementary and straight-forward approach.

References

D.M. Chickering (2002). Optimal structure identification with greedy search. Journal of Machine Learning Research 3, 507--554 A. Hauser and P. Bühlmann (2012). Characterization and greedy learning of interventional Markov equivalence classes of directed acyclic graphs. Journal of Machine Learning Research 13, 2409--2464.

See Also

gies, ges, Score, EssGraph

Examples

## Load predefined data
data(gmInt)

## Define the score (BIC)
score <- new("GaussL0penIntScore", gmInt$x, gmInt$targets, gmInt$target.index) 

## Estimate the essential graph
gds.fit <- gds(ncol(gmInt$x), gmInt$targets, score) 

## Plot the estimated essential graph and the true DAG
if (require(Rgraphviz)) {
  par(mfrow=c(1,2))
  plot(gds.fit$essgraph, main = "Estimated ess. graph")
  plot(gmInt$g, main = "True DAG")
}