Modeling Discrete Interventional Data using Directed ...

Modeling Discrete Interventional Data usingDirected Cyclic Graphical Models

Mark Schmidt and Kevin MurphyDepartment of Computer Science

University of British Columbia{schmidtm,murphyk}@cs.ubc.ca

Abstract

We outline a representation for discrete mul-tivariate distributions in terms of interven-tional potential functions that are globallynormalized. This representation can beused to model the effects of interventions,and the independence properties encoded inthis model can be represented as a directedgraph that allows cycles. In addition to dis-cussing inference and sampling with this rep-resentation, we give an exponential familyparametrization that allows parameter esti-mation to be stated as a convex optimiza-tion problem; we also give a convex relax-ation of the task of simultaneous parame-ter and structure learning using group `1-regularization. The model is evaluated onsimulated data and intracellular flow cytom-etry data.

1 Introduction

Graphical models provide a convenient framework forrepresenting independence properties of multivariatedistributions (Lauritzen, 1996). There has been sub-stantial recent interest in using graphical models tomodel data with interventions, that is, data wheresome of the variables are set experimentally. Directedacyclic graphical (DAG) models represent a joint dis-tribution over variables as a product of conditionalprobability functions, and are a convenient frame-work for modeling interventional data using Pearl’sdo-calculus (Pearl, 2000). However, the assumption ofacyclicity is often inappropriate; many models of bio-logical networks contain feedback cycles (for example,see Sachs et al. (2005)). In contrast, undirected graph-ical models represent a joint distribution over variablesas a globally normalized product of (unnormalized)clique potential functions, allowing cycles in the undi-

rected graph. However, the symmetry present in undi-rected models means that there is no natural notionof an intervention: For undirected models, there is nodifference between observing a variable (‘seeing’) andsetting it by intervention (‘doing’).

Motivated by the problem of using cyclic models forinterventional data, in this paper we examine a class ofdirected cyclic graphical models that represent a dis-crete joint distribution as a globally normalized prod-uct of (unnormalized) interventional potential func-tions, leading to a convenient framework for buildingcyclic models of interventional data. In §2, we reviewseveral highlights of the substantial literature on di-rected cyclic graphical models and the closely relatedtopic of representing distributions in terms of condi-tional functions. Subsequently, we discuss represent-ing a joint distribution over discrete variables with in-terventional potential functions (§3), the Markov in-dependence properties resulting from a graphical in-terpretation of these potentials (§4), modeling the ef-fects of interventions under this representation (§5),interpreting the model and interventions in the modelin terms of a data generating process involving feed-back (§6), inference and sampling in the model (§7),parameter estimation with an exponential family rep-resentation (§8), and a convex relaxation of structurelearning (§9). Our experimental results (§10) indicatethat this model offers an improvement in performanceover both directed and undirected models on both sim-ulated data and the data analyzed in (Sachs et al.,2005).

2 Related Work

Our work is closely related to a variety of previousmethods that express joint distributions in terms ofconditional distributions. For example, the classicwork on pseudo-likelihood for parameter estimation(Besag, 1975) considers optimizing the set of condi-tional distributions as a surrogate to optimizing the

joint distribution. Heckerman et al. (2000) have ad-vocated the advantages of dependency networks, di-rected cyclic models expressed in terms of conditionalprobability distributions (where ‘pseudo-Gibbs’ sam-pling is used to answer probabilistic queries). Theyargue that the set of conditional distributions may besimpler to specify than a joint distribution, and can becomputationally cheaper to fit. Closely related to de-pendency networks is the work of Hofmann and Tresp(1997), as well as work on conditionally specified dis-tributions (Arnold et al., 2001; Heckerman et al., 2004)(and the references contained in these works). How-ever, to our knowledge previous work on these modelshas not considered using globally normalized ‘condi-tional’ potential functions and trying to optimize thejoint distribution defined by their product, nor has itconsidered modeling the effects of interventions.

Our work is also closely related to work on pathdiagrams and structural equation models (SEMs)(Wright, 1921), models of functional dependence thathave long been used in genetics, econometrics, and thesocial sciences (see Pearl (2000)). Spirtes (1995) dis-cusses various aspects of ‘non-recursive’ SEMs, whichcan be used to represent directed cyclic feedback pro-cesses (as opposed to ‘recursive’ SEMs that can berepresented as a DAG). Spirtes (1995) shows that d-separation is a valid criterion for determining inde-pendencies from the graph structure in linear SEMs.Pearl and Dechter (1996) prove an analogous resultthat d-separation is valid for feedback systems involv-ing discrete variables. Modeling the effects of interven-tions in SEMs is discussed in (Strotz and Wold, 1960).Richardson (1996a,b) examines the problem of decid-ing Markov equivalence of directed cyclic graphicalmodels, and proposes a method to find the structure ofdirected cyclic graphs. Lacerda et al. (2008) recentlyproposed a new method of learning cyclic SEMs forcertain types of (non-interventional) continuous data.The representation described in this paper is distinctfrom this prior work on directed cyclic models in thatthe Markov properties are given by moralization of thedirected cyclic graph (§4), rather than d-separation.Further, we use potential functions to define a jointdistribution over the variables, while SEMs use deter-ministic functions to define the value of a child givenits parents (and error term).

A third thread of research related to this work isprior work on combinations of directed and undi-rected models. Modeling the effects of interventionsin chain graphs is thoroughly discussed in Lauritzenand Richardson (2002). Chain graphs are associatedwith yet another set of Markov properties and, unlikenon-recursive SEMs and our representation, requirethe restriction that the graph contains no partially di-

rected cycles. Also closely related are directed factorgraphs (Frey, 2003), but no interventional semanticshave been defined for these models.

3 Interventional PotentialRepresentation

We represent the joint distribution over a set of dis-crete variables xi (for i ∈ {1, . . . , n}) as a globallynormalized product of non-negative interventional po-tential functions

p(x1, . . . , xn) =1Z

n∏i=1

φ(xi|xπ(i)),

where π(i) is the set of ‘parents’ of node i, and thefunction φ(xi|xπ(i)) assigns a non-negative potentialto each joint configuration of xi and its parents xπ(i).The normalizing constant

Z =∑~x

∏i

φ(xi|xπ(i)),

enforces that the sum over all possible configurationsof ~x is unity. In contrast, undirected graphical modelsrepresent the joint distribution as a globally normal-ized product of non-negative potential functions de-fined on a set of C cliques,

p(x1, . . . , xn) =1Z

C∏c=1

φ(xc).

While in undirected graphical models we visualize thestructure in the model as an undirected graph withedges between variables in the same cliques, in the in-terventional potential representation we can visualizethe structure of the model as a directed graphG, whereG contains a directed edge going into each node fromeach of its parents. The global normalization allowsthe graph G defining these parent-child relationshipsto be an arbitrary directed graph between the nodes.

We obtain DAG models in the special case wherethe graph G is acyclic and for each node i thepotentials satisfy the local normalization constraint∀xπ(i)

∑xiφ(xi|xπ(i)) = 1. With these restrictions, the

interventional potentials represent conditional proba-bilities, and it can be shown that Z is constrained tobe 11. However, unlike DAG models, in our new rep-resentation the potentials do not need to satisfy any

1Because the global normalization makes the distri-bution invariant to re-scaling of the potentials, the dis-tribution will also be equivalent to a DAG model un-der the weaker condition that the graph is acyclic andfor each node i there exists a constant ci such that∀xπ(i)

∑xiφ(xi|xπ(i)) = ci. The conditional probability

functions in the corresponding DAG model are obtainedby dividing each potential by the appropriate ci.

Figure 1: The Markov blanket for node (T) includesits parents (P), children (C), and co-parents (Co). Thenode labeled C/P is both a child and a parent of T,and together they form a directed 2-cycle.

local normalization conditions, p(xi|xπ(i)) will not gen-erally be proportional to φ(xi|xπ(i)), and G is allowedto have directed cycles.

4 Markov Independence Properties

We define a node’s Markov blanket to be its parents,children, and co-parents (other parents of the node’schildren). If the potential functions are strictly posi-tive, then each node in the graph is independent of allother nodes given its Markov blanket:

p(xi|x−i) =p(xi, x−i)∑x′ip(x′i, x−i)

=

1Zφ(xi|xπ(i))

∏j 6=i,i/∈π(j)

φ(xj |xπ(j))∏

j 6=i,i∈π(j)

φ(xj |xπ(j))

∑x′i

1Zφ(x′i|xπ(i))

∏j 6=i,i/∈π(j)

φ(xj |xπ(j))∏

j 6=i,i∈π(j)

φ(xj |x′i, xπ(j)\i)

=

φ(xi|xπ(i))∏

j 6=i,i∈π(j)

φ(xj |xπ(j))∑x′i

φ(x′i|xπ(i))∏

j 6=i,i∈π(j)

φ(xj |x′i, xπ(j)\i)= p(xi|xMB(i)).

Above we have used x−i to denote all nodes exceptnode i, and xMB(i) to denote the nodes in the Markovblanket of node i. Figure 1 illustrates an example of anode’s Markov blanket.

In addition to this local Markov property, we can alsouse graphical operations to answer arbitrary queriesabout (conditional) independencies in the distribution.To do this, we first form an undirected graph by (i)placing an undirected edge between all co-parents that

are not directly connected, (ii) replacing all directed 2-cycles with a single undirected edge, and (iii) replacingall remaining directed edges with undirected edges. Totest whether a set of nodes P is independent of anotherset Q conditioned on a set R (denoted P ⊥ Q|R), it issufficient to test whether a path exists between a nodein P and a node in Q that does not pass through anynodes in R. If no such path exists, then the factoriza-tion of the joint distribution implies P ⊥ Q|R. Thisprocedure is closely related to the separation criterionfor determining independencies in undirected graphi-cal models (see Koller and Friedman (2009)), and itfollows from a similar argument that this test for in-dependence is sound2.

5 Effects of Interventions

Up to this point, the directed cyclic model can beviewed as a re-parameterization of an undirectedmodel. In this section we consider interventional data,which is naturally modeled using the interventional po-tential representation, but is not naturally modeled bythe (symmetric) clique potentials used in undirectedgraphical models.

In DAG models, we can incorporate an observationthat a variable xi takes on a specific value using therules of conditional probability (eg. p(x2:n|x1) =p(x1:n)/p(x1)). To model the effect of an intervention,where a variable xi is explicitly forced to take on a spe-cific value, we first remove the conditional mass func-tion p(xi|xπ(i)) from the joint probability, and then usethe rules of conditional probability on the resultingmodified distribution (Pearl, 2000). Viewed graphi-cally, removing the term from the joint distributiondeletes the edges going into the target of intervention,but preserves edges going out of the target.

By working with the interventional potential represen-tation, we can define the effect of setting a variableby intervention analogously to DAG models. Specifi-cally, setting a node xi by intervention corresponds toremoving the potential function φ(xi|xπ(i)) from thejoint distribution. The corresponding graphical oper-ation is similar to DAG models, in that edges goinginto targets of interventions are removed while edgesleaving targets are left intact3. In the case of directed

2For a specific set of interventional potential functions,there may be additional independence properties that arenot encoded in the graph structure (for example, if wehave deterministic dependencies or if we enforce the lo-cal normalization conditions needed for equivalence withDAG models). However, these are a result of the exact po-tentials used and will disappear under minor perturbationsof their values.

3The effect of interventions for SEMs is also analogous,in that an intervention replaces the structural equation for

Figure 2: Effects of interventions for a single directededge (top) and a directed 2-cycle (bottom). On the leftside we show the unmodified graph structure. On theright side, we show the effect on the graph structure ofintervening on the shaded node. For a single edge, theintervention leaves the graph unchanged if the parentis the target, and severs the edge if the child is thetarget. For the directed 2-cycle, we are left with asingle edge leaving the target.

2-cycles, the effect of an intervention is thus to changethe directed 2-cycle into a directed edge away from thetarget of intervention. Figure 2 illustrates the effectsof interventions in the case of a single edge, and in thecase of a directed 2-cycle. The independence prop-erties of the interventional distribution can be deter-mined graphically in the same way as the observational(non-interventional) distribution, by working with themodified graph. Note that intervention can not onlyaffect the independence properties between parent andchild nodes, but also between co-parents of the nodeset by intervention. Figure 3 gives an example.

These interventional semantics distinguish the inter-ventional potential representation of an undirectedmodel from the clique potential representation. Intu-itively, we can think of the directions in the graph asrepresenting undirected influences that are robust tointervention on the parent but not the child. That is, adirected edge from node i to j represents an undirectedstatistical dependency that remains after interventionon node i, but would not exist after intervention onnode j.

6 A Data Generating Process

Following §6 of Lauritzen and Richardson (2002), wecan consider a Markov chain Monte Carlo method forsimulating from a distribution represented with inter-

the target of intervention with a simple assignment oper-ator (Strotz and Wold, 1960). Graphically, this modifiesthe path diagram so that edges going into the target of in-tervention are removed but outgoing edges are preserved.

Figure 3: Independence properties in a simple graphbefore and after intervention on node T. From left toright we have (a) the original graph, (b) the undirectedgraph representing independence properties in the ob-servational distribution, (c) the modified graph afterintervention on T, and (d) the undirected graph repre-senting independence properties in the interventionaldistribution.

ventional potentials. In particular, consider the ran-dom Gibbs sampler where, beginning from some ini-tial ~x0, at each iteration we choose a node i at randomand sample xi according to p(xi|xMB(i)). If we stopthis algorithm after a sufficiently large number of it-erations that the Markov chain was able to convergeto its stationary (equilibrium) distribution, then thefinal value of ~x represents a sample from the distribu-tion. This data generating process involves feedbackin the sense that the value of each node is affected byall nodes connected (directly or indirectly) to it in thegraph. However, this process is different than previouscyclic feedback models in that the instantaneous valueof a variable is determined by its entire Markov blan-ket (as in undirected models), rather than its parentsalone (as in directed models).

The data generating process under conditioning (byobservation) sets the appropriate values of ~x0 to theirobserved values, and does not consider selecting ob-served nodes in the random update. The data gen-erating process under intervention similarly excludesupdating of nodes set by intervention. However, in-terventions can also affect the updating of nodes notset by intervention, since (i) a child set by interventionmay be removed from the Markov blanket, and/or (ii)co-parents of a child set by intervention may be re-moved from the Markov blanket. From this perspec-tive, we can give an interpretation to interventions inthe model; an intervention on a node i will remove ormodify (in the case of a directed 2-cycle) the instanta-neous statistical dependencies between node i and itsparents (and between co-parents of node i) in the equi-librium distribution. Of course, even if instantaneousstatistical dependencies are removed between a parentand child in the equilibrium distribution, the child setby intervention may still be able to indirectly affect its

parent in the equilibrium distribution if other pathsexist between the child and parent in the graph.

7 Inference and Sampling

When the total number of possible states is small, in-ference in the model can be carried out in a straightfor-ward way. Computing node marginals involves sum-ming the potentials for particular configurations anddividing by the normalizing constant

p(xi = c) =1Z

∑~x

Ic(xi)∏j

φ(xj |xπ(j)),

where Ic(xi) is the indicator function, taking a valueof 1 when xi takes the state c and 0 otherwise.

Computing marginals over the configurations of sev-eral nodes is performed similarly, while inference withobservations can be computed using the rules of con-ditional probability. We model interventions by re-moving the appropriate interventional potential func-tion(s) and computing a modified normalizing con-stant Z ′; the rules of conditional probability are thenused on the modified distribution. For example, if weintervene on node k, we can compute the marginal ofa node i (for i 6= k) using

p(xi = c|do(xk)) =1Z ′

∑x−k

Ic(xi)∏j 6=k

φ(xj |xπ(j)),

where the modified normalizing constant is

Z ′ =∑x−k

∏j 6=k

φ(xj |xπ(j)).

We can generate samples from the model using aninverse cumulative distribution function method; wegenerating a uniform deviate U in [0, 1], then computeeach term in the sum

∑~x

1Z

∏i φ(xi|xπ(i)) and stop at

the configuration where this sum first equals or sur-passes U .

For larger graphs, these computations are intractabledue to the need to compute the normalizing constant(and other sums over the set of possible configura-tions). Fortunately, the local Markov property willoften make it trivial to implement Gibbs sampling (orblock-Gibbs sampling) methods for the model (Gemanand Geman, 1984)4. It is also possible to take advan-

4Note that in general the entire Markov blanket isneeded to form the conditional distribution of a nodei. In particular, the ‘pseudo-Gibbs’ sampler where weloop through the nodes in some order and sample xi ∝φ(xi|xπ(i)) will not necessarily yield the appropriate sta-tionary distribution unless certain symmetry conditionsare satisfied (see Heckerman et al. (2000); Lauritzen andRichardson (2002)).

tage of dynamic programming methods for exact infer-ence (when the graph structure permits), and more so-phisticated variational and stochastic inference meth-ods (for example, see Koller and Friedman (2009)).However, a discussion of these methods is outside thescope of this paper.

8 Exponential Family ParameterEstimation

For a fixed graph structure, the maximum likelihoodestimate of the parameters given a data matrixX (con-taining m rows where each row is a sample of the nvariables) can be written as the minimization of thenegative log-likelihood function in terms of the param-eters θ of the interventional potentials.

− log p(X|θ) =−m∑d=1

n∑i=1

log(φ(Xd,i|Xd,π(i), θ))

+m logZ(θ).

If some elements of X are set by intervention, then theappropriate subset of the potentials and the modifiednormalizing constant must be used for these rows.

An appealing parameterization of the graphical modelis with interventional potential functions of the form

φ(xi|xπ(i), θ) = exp(bi,xi +∑

e∈{<i,j>:j∈π(i)}

wxi,xj ,e),

where each node i has a scalar bias bi,s for each dis-crete state s, and each edge e has a weight ws1,s2,e foreach state combination s1 and s2 (so θ is the union ofall bi,s and ws1,s2,e values). The gradient of the neg-ative log-likelihood with potentials in this form canbe expressed in terms of the training frequencies andmarginal probabilities as

−∇bi,s log p(X|θ) = −m∑d=1

Is(Xd,i)

+m p(xi = s|θ),

−∇ws1,s2,e log p(X|θ) = −m∑d=1

Is1(Xd,i)Is2(Xd,j)

+m p(xi = s1, xj = s2|θ).

Under this parameterization the joint distribution isin an exponential family form, implying that the neg-ative log-likelihood is a convex function in b and w.However, the exponential family representation willhave too many parameters to be identified from obser-vational data. For example, with observational datawe cannot uniquely determine the parameters in a di-rected 2-cycle. Even with interventional data the pa-rameters remain unidentifiable because, for example,

re-scaling an individual potential function does notchange the likelihood.

To make the parameters identifiable in our exper-iments, we perform MAP estimation with a small`2-regularizer added to the negative log-likelihood,transforming parameter estimation into a strictly con-vex optimization problem (this regularization also ad-dresses the problem that the unique infimum of thenegative log-likelihood may only be obtained with aninfinite value of some parameters, such as when wehave deterministic dependencies). Specifically, we con-sider the penalized log-likelihood

minθ− log p(X|θ) + λ2||θ||22,

where λ2 controls the scale of the regularizationstrength.

The dominant cost of parameter estimation is the cal-culation of the node and edge marginals in the gradi-ent. For parameter estimation in models where infer-ence is not tractable, the interventional potential rep-resentation suggests implementing a pseudo-likelihoodapproximation (Besag, 1975). Alternately, an approx-imate inference method could be used to compute ap-proximate marginals.

9 Convex Relaxation of StructureLearning

In many applications we may not know the appropriategraph structure. One way to write the problem ofsimultaneously estimating the parameters and graphstructure is with a cardinality penalty on the numberof edges E(G) for a graph structure G, leading to theoptimization problem

minθ,G− log p(X|θ) + λE(G),

where λ controls the strength of the penalty on thenumber of edges. We can relax the discontinuous car-dinality penalty (and avoid searching over graph struc-tures) by replacing the second term with a group `1-regularizer (Yuan and Lin, 2006) on appropriate ele-ments of w (each group is the elements w.,.,e associatedwith edge e), giving the problem

minθ− log p(X|θ) + λ

∑e

||w.,.,e||2, (1)

where θ = {b.,., w.,.,.}. Solving this continuous opti-mization problem for sufficiently large λ yields a sparsestructure, since setting all values w.,.,.e to zero for aparticular edge e is equivalent to removing the edgefrom the graph. This type of approach to simultaneousparameter and structure learning has previously been

explored for undirected graphs (see Lee et al., 2006;Schmidt et al., 2008), but can not be used directly forDAG models (unless we restrict ourselves to a fixednode ordering) because of the acyclicity constraint.

Similar to Schmidt et al. (2008), we can convert thecontinuous and unconstrained but non-differentiableproblem (1) into a differentiable problem with second-order cone constraints

minθ− log p(X|θ) + λ

∑e

αe,

s.t. αe ≥ ||w.,.,e||2.

Since the edge parameters form disjoint sets, projec-tion onto these constraints is a trivial computation(Boyd and Vandenberghe, 2004, Exercise 8.3(c)), andthe optimization problem can be efficiently solved us-ing a limited-memory projected quasi-Newton method(Schmidt et al., 2009)

10 Experiments

We compared the performance of several differentgraphical model representations on two data sets. Theparticular models we compared were:

• DAG: A directed acyclic graphical model trainedwith group `1-regularization on the edges. Weused the projected quasi-Newton method ofSchmidt et al. (2009) to optimize the criteriafor a given ordering, and used the dynamic pro-gramming algorithm of Silander and Myllymaki(2006) to minimize the regularized negative log-likelihood over all possible node orderings. Theinterventions are modeled as described in §5.

• UG-observe: An undirected graphical modeltrained with group `1-regularization on the edgesthat treats the data as if it was purely ob-servational. Specifically, it seeks to maximizep(x1, . . . , xn) over the training examples (subjectto the regularization) and ignores that some of thenodes were set by intervention.

• UG-condition: An undirected graphical modeltrained with group `1-regularization on the edgesthat conditions on nodes set by intervention.Specifically, it seeks to maximize p(x1, . . . , xn)over the training examples (subject to the regu-larization) on observational samples, and seeks tomaximize p(x−k|xk) over the training examples(subject to the regularization) on interventionalsamples where node k was set by intervention.

• DCG: The proposed directed cyclic graphicalmodel trained with group `1-regularization, mod-eling the interventions as described in §5.

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Figure 4: Results on data generated from 4 different DCG models.

To make the comparisons fair, we used a linear expo-nential family representation for all models. Specifi-cally, the DAG model uses conditional probabilities ofthe form

p(xi|xπ(i), θ) =1Zi

exp(bi,xi +∑j∈π(i)

wxi,xj ,e),

(where Zi normalizes locally), while the UG modelsuse a distribution of the form

p(x1, . . . , xn|θ) =1Z

exp(n∑i=1

bi,xi+∑

e:{<i,j>∈E}

wxi,xj ,e),

and the DCG model uses the interventional potentialfunctions described in §8.

To ensure identifiability of all model parametersand increase numerical stability, we applied `2-regularization to the parameters of all models. Weset the scale λ2 of the `2-regularization parameter to10−4, but our experiments were not particularly sen-sitive to this choice. The groups used in all methodswere simply the set of parameters associated with anindividual edge.

10.1 Directed Cyclic Data

We first compared the performance in terms of test-setnegative log-likelihood on data generated from an in-terventional potential model. We generated the graphstructure by including each possible directed edge withprobability 0.5, and sampled the node and edge param-eters from a standard normal distribution, N (0, 1).We generated 1000 samples from a 10-node binarymodel, where in 1/11 of the samples we generated apurely observational sample, and in 10/11 of the sam-ples we randomly choose one of the ten nodes and setit by intervention. We repeated this 10 times to gen-erate 10 different graph structures and parameteriza-tions, and for each of these we trained on the first500 samples and tested on the remaining 500. In Fig-ure 4, we plot the test set negative log-likelihood (ofnodes not set by intervention) against the strength ofthe group `1-regularization parameter for the first 4of these trials (the others yielded qualitatively similarresults).

We first contrast the performance of the DAG model(which models the effects of intervention but can notmodel cycles) with the UG-condition method (whichdoes not model the effects of intervention but canmodel cycles). In our experiments, neither of these

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Figure 5: Mean results (plus/minus two standard devi-ations) on the expression data from Sachs et al. (2005)over 10 training/test splits.

methods dominated the other; in most distributionsthe optimal DAG had a small advantage over UG-condition for suitably chosen values of the regulariza-tion parameter, while in other experiments the UG-condition model offered a small advantage. In con-trast, the DCG model (which models the effects of in-terventions and also allows cycles) outperformed boththe DAG and the UG methods over all 10 experiments.Finally, in terms of the two UG models, conditioningon the interventions during training strictly dominatedtreating the data as observational, and in all cases theUG-observe model was the worst among the 4 meth-ods.

10.2 Cell Signaling Network

We next applied the methods to the data studiedin Sachs et al. (2005). In this study, intracellularmultivariate flow cytometry was used to simultane-ously measure the expression levels of 11 phophory-lated proteins and phospholipid components in indi-vidual primary human immune system cells under 9different stimulatory/inhibitory conditions. The dataproduced in this work is particularly amenable to sta-tistical analysis of the underlying system, because in-tracelleular mulitvariate flow cytomery allows simul-taneous measurement of multiple proteins states inindividual cells, yielding hundreds of data points foreach interventional scenario. In Sachs et al. (2005),a multiple restart simulated annealing method wasused to search the space of DAGs, and the final graphstructure was produced by averaging over a set of themost high-scoring networks. Although this methodcorrectly identified many edges that are well estab-lished in the literature, the method also missed threewell-established connections. The authors hypothe-sized that the acyclicity constraint may be the reason

that the edges were missed, since they could have in-troduced directed cycles (Sachs et al., 2005). In princi-ple, these edges could be discovered using DAG modelsof time-series data. However, current technology doesnot allow collection of this type of data (Sachs et al.,2005), motivating the need to examine cyclic modelsof interventional data.

In our experiments, we used the targets of interven-tion and 3-state discretization strategy (into ‘under-expressed’, ‘baseline’, and ‘over-expressed’) of Sachset al. (2005). We trained on 2700 randomly chosensamples and tested on the remaining 2700, and re-peated this on 10 other random splits to assess thevariability of the results. Figure 5 plots the mean testset likelihood (and two standard deviations) across the10 trials for the different methods. On this real dataset, we see similar trends to the synthetic data sets. Inparticular, the UG-observe model is again the worst,the UG-condition and DAG models have similar per-formance, while the DCG model again dominates bothDAG and UG methods.

Despite its improvement in predictive performance,the learned DCG graph structures are less inter-pretable than previous models. In particular, thegraphs contain a large number of edges even for smallvalues of the regularization parameter (this may bedue to the use of the linear parameterization). Thegraphs also include many directed cycles, and forlarger values of λ incorporate colliders whose parentsdo not share an edge. It is interesting that the (pair-wise) potentials learned for 2-cycles in the DCG modelwere often very asymmetric.

11 Discussion

While we have assumed that the interventions are ‘per-fect’, in many cases it might be more appropriate touse DCG models with ‘imperfect’, ‘soft’, or ‘uncer-tain’ interventions (see Eaton and Murphy (2007)).We have also assumed that each edge is affected asym-metrically by intervention, while we could also considerundirected edges that are affected symmetrically. Forexample, we could consider ‘stable’ undirected edgesthat represent fundamentally associative relationshipsthat remain after intervention on either target. Alter-nately, we could consider ‘unstable’ undirected edgesthat are removed after intervention on either target(this would be appropriate in the case of a hidden com-mon cause).

To summarize, the main contribution of this work is amodel for interventional data that allows cycles. Itis therefore advantageous over undirected graphicalmodels since it offers the possibility to distinguish be-tween ‘seeing’ and ‘doing’. However, unlike DAG mod-

els that offer the same possibility, it allows cycles inthe model and thus may be more well-suited for datasets like those generated from biological systems whichoften have natural cyclic behaviour.

Acknowledgements

We would like to thanks the anonymous reviewers forhelpful suggestions that improved the paper.

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