Active learning of causal networks with intervention experiments and optimal designs

Abstract

The causal discovery from data is important for various scientific investigations. Because we cannot distinguish the different directed acyclic graphs (DAGs) in a Markov equivalence class learned from observational data, we have to collect further information on causal structures from experiments with external interventions. In this paper, we propose an active learning approach for discovering causal structures in which we first find a Markov equivalence class from observational data, and then we orient undirected edges in every chain component via intervention experiments separately. In the experiments, some variables are manipulated through external interventions. We discuss two kinds of intervention experiments, randomized experiment and quasi-experiment. Furthermore, we give two optimal designs of experiments, a batch-intervention design and a sequential-intervention design, to minimize the number of manipulated variables and the set of candidate structures based on the minimax and the maximum entropy criteria. We show theoretically that structural learning can be done locally in subgraphs of chain components without need of checking illegal v-structures and cycles in the whole network and that a Markov equivalence subclass obtained after each inter-vention can still be depicted as a chain graph.

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Journal of Machine Learning Research 9 (2008) 2523-2547Submitted 6/07; Revised 2/08; Published 11/08
Active Learning of Causal Networks
with Intervention Experiments and Optimal Designs
Yang-Bo He
Zhi Geng
School of Mathematical Sciences, LMAM
Peking University
Beijing 100871, China
HEYB@MATH.PKU.EDU.CN
ZGENG@MATH.PKU.EDU.CN
Editor: Andre Elisseeff
Abstract
Thecausal discoveryfromdataisimportantforvariousscientificinvestigations. Becausewecannot
distinguishthedifferentdirectedacyclicgraphs(DAGs)inaMarkovequivalence classlearnedfrom
observational data, we have to collect further information on causal structures from experiments
with external interventions. In this paper, we propose an active learning approach for discovering
causal structures in which we first find a Markov equivalence class from observational data, and
then we orient undirected edges in every chain component via intervention experiments separately.
In the experiments, some variables are manipulated through external interventions. We discuss two
kinds of intervention experiments, randomized experiment and quasi-experiment. Furthermore, we
give two optimal designs of experiments, a batch-intervention design and a sequential-intervention
design, to minimize the number of manipulated variables and the set of candidate structures based
on the minimax and the maximum entropy criteria. We show theoretically that structural learning
can be done locally in subgraphs of chain components without need of checking illegal v-structures
and cycles in the whole network and that a Markov equivalence subclass obtained after each inter-
vention can still be depicted as a chain graph.
Keywords: active learning, causal networks, directed acyclic graphs, intervention, Markov equiv-
alence class, optimal design, structural learning
1. Introduction
A directed acyclic graph (DAG) (also called a Bayesian network) is a powerful tool to describe
a large complex system in various scientific investigations, such as bioinformatics, epidemiology,
sociology and business (Pearl, 1988; Lauritzen, 1996; Whittaker, 1990; Aliferis et al., 2003; Jansen
et al., 2003; Friedman, 2004). A DAG is also used to describe causal relationships among variables.
It is crucial to discover the structure of a DAG for understanding a large complex system or for
doing uncertainty inference on it (Cooper and Yoo, 1999; Pearl, 2000). There are many methods of
structural learning, and the main methods are Bayesian methods (Cooper and Yoo, 1999; Hecker-
man, 1997) and constraint-based methods (Spirtes et al., 2000). From data obtained in observational
studies, we may not have enough information to discover causal structures completely, but we can
obtain only a Markov equivalence class. Thus we have to collect further information of causal
structures via experiments with external interventions. Heckerman et al. (1995) discussed structural
learning of Bayesian networks from a combination of prior knowledge and statistical data. Cooper
and Yoo (1999) presented a method of causal discovery from a mixture of experimental and obser-
c ?2008 Yang-Bo He and Zhi Geng.

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HE AND GENG
vational data. Tian and Pearl (2001a,b) proposed a method of discovering causal structures based
on dynamic environment. Tong and Koller (2001) and Murphy (2001) discussed active learning
of Bayesian network structures with posterior distributions of structures based on decision theory.
In these methods, causal structures are discovered by using additional information from domain
experts or experimental data.
Chain graphs were introduced as a natural generalization of DAGs to admit more flexible causal
interpretation (Lauritzen and Richardson, 2002). A chain graph contains both directed and undi-
rected edges. A chain component of a chain graph is a connected undirected graph obtained by
removing all directed edges from the chain graph. Andersson et al. (1997) showed that DAGs in a
Markov equivalence class can be represented by a chain graph. He et al. (2005) presented an ap-
proach of structural learning in which a Markov equivalence class of DAGs is sequentially refined
into some smaller subclasses via domain knowledge and randomized experiments.
In this paper, we discuss randomized experiments and quasi-experiments of external interven-
tions. We propose a method of local orientations in every chain component, and we show theo-
retically that the method of local orientations does not create any new v-structure or cycle in the
whole DAG provided that neither v-structure nor cycle is created in any chain component. Thus
structural learning can be done locally in every chain component without need of checking illegal
v-structures and cycles in the whole network. Then we propose the optimal designs of interventional
experiments based on the minimax and maximum entropy criteria. These results greatly extend the
approach proposed by He et al. (2005). In active learning, we first find a Markov equivalence class
from observational data, which can be represented by a chain graph, and then we orient undirected
edges via intervention experiments. Two kinds of intervention experiments can be used for orien-
tations. One is randomized experiment, in which an individual is randomly assigned to some level
combination of the manipulated variables at a given probability. Randomization can disconnect the
manipulated variables from their parent variables in the DAG. Although randomized experiments
are most powerful for learning causality, they may be inhibitive in practice. The other is quasi-
experiment, in which the pre-intervention distributions of some variables are changed via external
interventions, but we cannot ensure that the manipulated variables can be disconnected from their
parent variables in the DAG, and thus the post-intervention distributions of manipulated variables
may still depend on their parent variables. For example, the pre-intervention distribution of whether
patients take a vaccine or not may depend on some variables, and the distribution may be changed
by encouraging patients with some benefit in the quasi-experiment, but it may still depend on these
variables. Furthermore, we discuss the optimal designs by which the number of manipulated vari-
ables is minimized or the uncertainty of candidate structures is minimized at each experiment step
based on the minimax and the maximum entropy criteria. We propose two kinds of optimal designs:
a batch-intervention experiment and a sequential intervention experiment. For the former, we try to
find the minimum set of variables to be manipulated in a batch such that undirected edges are all
oriented after the interventions. For the latter, we first choose a variable to be manipulated such that
the Markov equivalence class can be reduced by manipulating the variable into a subclass as small
as possible, and then according to the current subclass, we repeatedly choose a next variable to be
manipulated until all undirected edges are oriented.
In Section 2, we introduce notation and definitions and then show some theoretical results on
Markov equivalence classes. In Section 3, we present active learning of causal structures via ex-
ternal interventions and discuss randomized experiments and quasi-experiments. In Section 4, we
propose two optimal designs of intervention experiments, a batch-intervention design and a sequen-
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tial intervention design. In Section 5, we show simulation results to evaluate the performances of
intervention designs proposed in this paper. Conclusions are given in Section 6. Proofs of theorems
are given in Appendix.
2. Causal DAGs and Markov Equivalence Class
A graph G can be defined to be a pair G = (V,E), where V = {V1,···,Vn} denotes the node set and
E denotes the edge set which is a subset of the set V×V of ordered pairs of nodes. If both ordered
pairs (Vi,Vj) and (Vj,Vi) are in E, we say that there is an undirected edge betweenViandVj, denoted
as Vi−Vj. If (Vi,Vj) ∈ E and (Vj,Vi) / ∈ E, we call it a directed edge, denoted as Vi→Vj. We say
that Viis a neighbor of Vjif there is an undirected or directed edge between Viand Vj. A graph is
directed if all edges of the graph are directed. A graph is undirected if all edges of the graph are
undirected.
A sequence (V1,V2,···,Vk) is called a partially directed path fromV1toVkif eitherVi→Vi+1or
Vi−Vi+1is in G for all i = 1,...,k−1. A partially directed path is a directed path if there is not any
undirected edge in the path. A node Viis an ancestor of Vjand Vjis a descendant of Viif there is a
directed path from Vito Vj. A directed cycle is a directed path from a node to itself, and a partially
directed cycle is a partially directed path from a node to itself.
A graph with both directed and undirected edges is a chain graph if there is not any partially
directed cycle. Figure 1 shows a chain graph with five nodes. A chain component is a node set
whose nodes are connected in an undirected graph obtained by removing all directed edges from
the chain graph. An undirected graph is chordal if every cycle of length larger than or equal to 4
possesses a chord.
r
r
r
r
r
V1
QQQ
V2
V3
V4
V5
?
?
?
QQQQQQQ
?
s
Figure 1: A chain graph G∗depicts the essential graph of G,G1,G2and G3.
A directed acyclic graph (DAG) is a directed graph which does not contain any directed cycle. A
causal DAG is a DAG which is used to describe the causal relationships among variablesV1,···,Vn.
In the causal DAG, a directed edgeVi→Vjis interpreted as that the parent nodeViis a cause of the
child node Vj, and that Vjis an effect of Vi. Let pa(Vi) denote the set of all parents of Viand ch(Vi)
denote the set of all children ofVi. Let τ be a node subset of V. The subgraph Gτ= (τ,Eτ) induced
by the subset τ has the node set τ and the edge set Eτ= E∩(τ×τ) which contains all edges falling
into τ. Two graphs have the same skeleton if they have the same set of nodes and the same set of
edges regardless of their directions. A head-to-head structure is called a v-structure if the parents
are not adjacent, such asV1→V2←V3.
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HE AND GENG
Figure 2 shows four different causal structures of five nodes. The causal graph G in Figure 2
depicts thatV1is a cause ofV3, which in turn is a cause ofV5.
r
V2
V3
QQQQQQ
s
V4
V5
s
r
r
r
r
V1
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V4
V5
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G
r
r
r
r
r
V1
V2
V3
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3QQQ
s
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G1
r
r
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V1
V2
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6
3QQQ
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G2
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Qk
V2
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V5
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G3
Figure 2: The equivalence class [G].
A joint distribution P satisfies Markov property with respect to a graph G if any variable of G is
independent of all its non-descendants in G given its parents with respect to the joint distribution P.
Furthermore, the distribution P can be factored as follows
P(v1,v2,···,vn) =
n
∏
i=1
P(vi|pa(vi)),
where videnotes a value of variable Vi, and pa(vi) denotes a value of the parent set pa(Vi) (Pearl,
1988; Lauritzen, 1996; Spirtes et al., 2000). In this paper, we assume that any conditional indepen-
dence relations in P are entailed by the Markov property, which is called the faithfulness assump-
tion (Spirtes et al., 2000). We also assume that there are no latent variables (that is, no unmeasured
variables) in causal DAGs. Different DAGs may encode the same Markov properties. A Markov
equivalence class is a set of DAGs that have the same Markov properties. Let G1∼ G2denote that
two DAGs G1and G2are Markov equivalent, and let [G] denote the equivalence class of a DAG
G, that is, [G] = {G?: G?∼ G}. The four DAGs G, G1, G2and G3in Figure 2 form a Markov
equivalence class [G]. Below we review two results about Markov equivalence of DAGs given by
Verma and Pearl (1990) and Andersson et al. (1997).
Lemma 1 (Verma and Pearl, 1990) Two DAGs are Markov equivalent if and only if they have the
same skeleton and the same v-structures.
Andersson et al. (1997) used an essential graph G∗to represent the equivalence class [G].
Definition 2 The essential graph G∗= (V,E∗) of G has the same node set and the same skeleton
as G, whose one edge is directed if and only if it has the same orientation in every DAG in [G] and
whose other edges are undirected.
For example, G∗in Figure 1 is the essential graph of G in Figure 2. The edges V2→ V5and
V3→V5in G∗are directed since they have the same orientation for all DAGs of [G] in Figure 2, and
other edges are undirected.
Lemma 3 (Andersson et al., 1997) Let G∗be the essential graph of G = (V,E). Then G∗has the
following properties:
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(i) G∗is a chain graph,
(ii) G∗
τis chordal for every chain component τ, and
(iii) Vi→Vj−Vkdoes not occur as an induced subgraph of G∗.
Suppose that G is an unknown underlying causal graph and that its essential graph G∗= (V,E)
has been obtained from observational data, and has k chain components {τ1,···,τk}. Its edge set
E can be partitioned into the set E1of directed edges and the set E2of undirected edges. Let G∗
denote a subgraph of the essential G∗induced by a chain component τ of G∗. Any subgraph of the
essential graph induced by a chain component is undirected. Since all v-structures can be discovered
from observational data, any subgraph G?τof G?should not have any v-structure for G?∈ [G]. For
example, the essential graph G∗in Figure 1 has one chain component τ = {V1,V2,V3,V4}. It can
been seen that G?τhas no v-structure for G?∈ {G,G1,G2,G3}.
Given an essential graph G∗, we need to orient all undirected edges in each chain component
to discover the whole causal graph G. Below we show that the orientation can be done separately
in every chain component. We also show that there are neither new v-structures nor cycles in the
whole graph as long as there are neither v-structures nor cycles in any chain component. Thus in
the orientation process, we only need to ensure neither v-structures nor cycles in any component,
and we need not check new v-structures and cycles for the whole graph.
τ
Theorem 4 Let τ be a chain component of an essential graph G∗. For each undirected edgeV −U
in G∗
that is, neitherV →U ←W norW →V ←U can occur for anyW / ∈ τ.
Theorem 4 means that there is not any node W outside the component τ which can build a
v-structure with two nodes in τ.
τ, neither orientation V →U nor V ←U can create a v-structure with any node W outside τ,
Theorem 5 Let τ be a chain component of G∗. If orientation of undirected edges in the subgraph
G∗
directed cycle in the whole DAG.
τdoes not create any directed cycle in the subgraph, then the orientation does not create any
According to Theorems 4 and 5, we find that the undirected edges can be oriented separately
in each chain component regardless of directed and undirected edges in other part of the essential
graph as long as neither cycles nor v-structures are constructed in any chain component. Thus the
orientation for one chain component does not affect the orientations for other components. The
orientation approach and its correctness will be discussed in Section 3.
3. Active Learning of Causal Structures via External Interventions
To discover causal structures further from a Markov equivalence class obtained from observational
data, we have to perform external interventions on some variables. In this section, we consider two
kinds of external interventions. One is the randomized experiment, in which the post-intervention
distribution of the manipulated variable Viis independent of its parent variables. The other is the
quasi-experiment, in which the distribution of the manipulated variableViconditional on its parents
pa(Vi) is changed by manipulating Vi. For example, the distribution of whether patients take a
vaccine or not is changed by randomly encouraging patients at a discount.
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