Interventions, Where and How? Bayesian Active Causal Discovery at Scale
Anonymous Author(s)
AfﬁliationAddress email
Abstract
Causal discovery from observational and interventional data is challenging dueto limited data and non-identiﬁability: factors that introduce uncertainty in estimating the underlying structural causal model (SCM).
Selecting experiments(interventions) based on the uncertainty arising from both factors can expeditethe identiﬁcation of the SCM.
Existing methods in experimental design for causaldiscovery from limited data either rely on linear assumptions for the SCM or selectonly the intervention target.
This work incorporates recent advances in Bayesiancausal discovery into the Bayesian optimal experimental design framework, allowing for active causal discovery of large, nonlinear SCMs while selecting boththe interventional target and the value.
We demonstrate the performance of theproposed method on synthetic graphs (Erdos-Rènyi, Scale Free) for both linearand nonlinear SCMs as well as on the in-silico single-cell gene regulatory networkdataset, DREAM.
Introduction
[Figure]
What is the structure of the protein-signaling networkderived from a single cell?
How do different habitsinﬂuence the presence of disease?
Such questionsrefer to causal effects in complex systems governedby nonlinear, noisy processes.
On most occasions,passive observation of such systems is insufﬁcient touncover the real cause-effect relationship and costlyexperimentation is required to disambiguate betweencompeting hypotheses.
As such, the design of experiments is of signiﬁcant interest; an efﬁcient experimentation protocol helps reduce the costs involved in experimentation while aiding the process of producingknowledge through the (closed-loop / policy-driven)scientiﬁc method (Fig. 1).
In the language of causality [32], the causal relationships are represented qualitatively by a directedacyclic graph (DAG), where the nodes correspond todifferent variables of the system of study and the edges represent the ﬂow of information between thevariables.
The abstraction of DAGs allows us to represent the space of possible explanations (hypotheses) for the observations at hand.
Representing such hypotheses as Bayesian probabilities (beliefs)allows us to formalize the problem of the scientiﬁc method as one of Bayesian inference, where thegoal is to estimate the posterior distribution p ( DAGs | Observations ) .
A posterior distribution over the DAGs allows us to employ information-theoretic acquisition functions that guide experimentationtowards the most informative variables for disambiguating between competing hypotheses.
Suchdesign procedures belong to the ﬁeld of Bayesian Optimal Experimental Design [24] for Causal
Discovery (BOECD) [48, 29].
In the Bayesian Optimal Experimental Design (BOED) [24] framework, one seeks the experiment thatmaximizes the expected information gain about some parameter(s) of interest.
In causal discovery ,an experiment takes the form of a causal intervention, and the parameters of interest are the DAGstructure and the associated Structural Causal Model (SCM) consisting of a set of structural equationsrepresenting the functional relationships between the variables in the DAG.
An intervention in a causal model refers to the variable (or target) we manipulate and the value (orstrength) at which we set the variable.
Hence, the design space in the case of learning causal models isthe set of all subsets of the intervention targets and the possibly countably inﬁnite set of interventionvalues of the chosen targets.
The intervention value encapsulates important semantics in many causalinference applications.
For instance, in medical applications, an intervention can correspond to theadministration of different drugs and the intervention value takes the form of a dosage level for eachdrug.
Even though the appropriate choice of this value is crucial for identifying the underlying causalmodel, existing work on active causal discovery focuses exclusively on selecting the interventiontarget [1, 8].
There, the intervention value is generally some arbitrary ﬁxed value (like 0) whichis suboptimal (see Fig. 2a).
Hence, a holistic treatment of selecting the intervention value and thetarget in the general case of nonlinear causal models has been missing.
We present a simple yeteffective causal Bayesian experimental design method ( CBED - pronounced “seabed”) to acquireoptimal intervention targets and values by performing Bayesian optimization.
Additionally, some settings call for the selection of a batch of interventions.
The problem of batchedinterventions is computationally expensive as it requires evaluating all possible combinations ofinterventions.
We extend CBED to the batch setting and propose two different batching strategiesfor tractable, Bayes optimal acquisition of both intervention targets and values.
The ﬁrst strategy —
Greedy-CBED — builds up the intervention set greedily.
A greedy heuristic is still near-optimal dueto submodularity properties of mutual information [22, 1, 19].
The second strategy — Soft-CBED— constructs a set of interventions by stochastic sampling from a ﬁnite set of candidates, therebysigniﬁcantly increasing computational efﬁciency while recovering the causal graph and learningthe parameters of the SCM as fast as the greedy strategy.
This strategy is well suited for resourceconstrained settings.
Throughout this work, we make the following standard assumptions for causal discovery [36]:
Assumption 1 (Causal Sufﬁciency) .
There are no hidden confounders, and all the random variablesof interest are observable.
Assumption 2 (Finite Samples) .
There is a ﬁnite number of observational/ interventional samplesavailable.
Assumption 3 (Nonlinear SCM with Additive Noise) .
The structural causal model has nonlinearconditional expectations with additive Gaussian noise.
Assumption 4 (Single Target) .
Each intervention is atomic and applied to a single target of the
SCM.
Additionally, we assume that interventions are planned and executed in batches of size B , with a ﬁxedbudget of total interventions given by Number of Batches ⇥ B .
We also assume that the underlyinggraph is sparse, as is the case in all the real-world settings [5, 42].
Finally, we are interested inrecovering the full graph G with a small number of batches.
We show that our methods, Greedy-CBED and Soft-CBED , perform better than the state-of-the-artactive causal discovery baselines in linear and nonlinear SCM settings.
In addition, our approachachieves superior results in the real-world inspired nonlinear dataset, DREAM [15].
Background
Notation.
Let V = { 1 , . . .
, d } be the vertex set of any DAG g =
( V , E ) and X V =
{ X 1 , . .
, X d } ✓ X be the random variables of interest indexed by V .
We have an initial observationaldataset D
= { x V ( i ) } ni =1 comprised of instances x V ⇠ P (X 1 = x 1 , . .
, X d = x d ) =
p (x 1 , . . . , x d ) .
Causal Bayesian Network.
A causal Bayesian network (CBN) is the pair ( g , P ) such that for any
[Equation]
⇢
[Equation] [Equation]
where do( X W ) represents a hypothetical intervention on the variables X W , 1 ( · ) is an indicatorfunction and pa g ( i ) denotes parents of variable X i in DAG g .
A perfect intervention on any variable
X j completely removes all dependencies with its parents, i.e. P ( X j | X pa g ( j ) )
=
P ( X j ) therebyresulting in a mutilated DAG g
= ( V , E \ ( pa g ( j ) , j )) .
Structural Causal Model.
From the data generative mechanism point of view, the DAG g on X Vmatches a set of structural equations :
[Equation]
where f i ’s are (potentially nonlinear) causal mechanisms that remain invariant when interveningon any variable X j 6 =
X i .
✏ i ’s are exogenous noise variables with arbitrary distribution that aremutually independent, i.e ✏ i ??
✏ j 8 i 6 = j .
(1) represents the conditional distributions in a Causal 99 Bayesian Network and can additionally reveal the effect of interventions if the mechanisms areknown [36, 32].
These equations, along with the DAG g , are known as the structural causal model(SCM).
Though the mechanisms f can be nonparametric in the general case, we assume that thereexists a parametric approximation to these mechanisms with parameters  2  .
In the case of linear SCMs,  corresponds to the weights of the edges in E .
In the nonlinear case, they could representthe parameters of a nonlinear function that parameterizes the mean of a Gaussian distribution.
A common form of (1) corresponds to Gaussian additive noise models (ANM) 1 :
[Equation]
An ANM is fully speciﬁed by a a DAG, g , mechanisms, f ( · ;  )= [ f 1 ( · ;  1 ) , . . , f d ( · ;  d )] ,parameterized by  =
[  1 , . . .
,  d ] , and variances,  2 = ⇥  21 , . . .
,  2 d ⇤ .
For notational brevity,henceforth we denote ✓ =
(  ,  2 ) and all the parameters of the SCM with  =
( g , ✓ ) .
Bayesian Causal Discovery.
A common assumption in causal inference is that causal relationsare known qualitatively and can be represented by a DAG.
While this qualitative information canbe obtained from domain knowledge in some scenarios, it’s infeasible in most applications.
Thegoal of causal discovery is to recover the SCM given a dataset D .
In general, without furtherassumptions about the nature of mechanisms f (e.g., linear vs. nonlinear), the true SCM may notbe identiﬁable [34] from observational data alone.
This non-identiﬁability is because there couldbe multiple DAGs (and hence multiple factorizations of P ( X V ) ) which explain the data equallywell.
Such DAGs are said to be Markov Equivalent .
Interventions can improve identiﬁability.
Inaddition to identiﬁability issues, estimating the functional relationships between nodes using ﬁnitedata is another source of uncertainty.
Bayesian parameter estimation over the unknown SCM providesa principled way to quantify these uncertainties and obtain a posterior distribution over the SCMgiven observational data.
An experimenter can then use the knowledge encoded by the posterior todesign informative experiments that efﬁciently acquire interventional data to resolve unknown edgeorientations and functional uncertainty .
Posterior Distributions over SCMs.
The key challenge in obtaining posteriors over SCMs isthat the space of DAGs is discrete and superexponential in the number of variables [36].
However,recent techniques based on variational inference [2, 26, 9] provide a tractable and scalable way ofapproximating this posterior.
Given a tractable approximation of the posterior over the parameters , variational inference optimizes a lower bound on the (log-) evidence:
[Equation]
The key idea in these techniques is the way the variational family for SCMs, in particularover DAGs, is parameterized.
The variational family for the Variational Causal Network (VCN) method [2] is an autoregressive Bernoulli distribution over the adjacency matrix.
They furtherenforce the acyclicity constraint [51] through the prior.
BCD-Nets [9] consider a distribution overnode orderings through a Boltzmann distribution and perform inference with Gumbel-Sinkhorn [27]operator.
DiBS [26] consider latent variables over entries of adjacency matrix and perform inferenceover these latent variables using SVGD [25].
We demonstrate empirical results in BOECD usingthe DiBS model in this work because it is easily extendable to nonlinear SCMs.
Bayesian Optimal Experimental Design.
Bayesian Optimal Experimental Design (BOED) [24, 7]is an information theoretic approach to the problem of selecting the optimal experiment to estimateany parameter ✓ .
For BOED, the utility of the experiment ⇠ is the mutual information (MI) betweenthe observation y and ✓ :
[Equation]
The goal of BOED is to select the experiment that maximizes this objective ⇠ ⇤ = arg max ⇠ U BOED ( ⇠ ) .
A common setting, called static , ﬁxed or batch design, is to optimize multiple designs { ⇠ 1 , . . .
, ⇠ B }at the same time.
Method
The true SCM ˜  =
(˜ g , ˜ ✓ ) over random variables
X V is a matter of fact, but our belief in ˜  isuncertain for many reasons.
Primarily, it is only possible to learn the DAG ˜
g up to a Markovequivalence class (MEC) from observational data D .
Uncertainty also arises from D from beinga ﬁnite sample, which we model by introducing the the random variable  , of which  is an outcome.
Let  ⇠ p (  |D ) / p ( D |  ) p (  ) be an instance of the random variable  that is sampled from 149 our posterior over SCMs after observing the dataset
D .
We would like to design an experiment to identify an intervention ⇠ := { ( j, v ) } :=
do(X j = v ) thatmaximizes the information gain about  after observing the outcome of the intervention y ⇠ P (X 1 =x 1 , . . .
, X d = x d | do(X j = v ) =
p ( y | ⇠ ) .
Here, y is an instance of the random variable Y ✓ Xdistributed according to the distribution speciﬁed by the mutilated true graph ˜ g 0 under intervention.
Looking at one intervention at a time, one can formalize BOECD as gain in information about after observing the outcome of an experiment y .
The experiment ⇠ := { ( j, v ) } that maximizes theinformation gain is the experiment that maximizes the mutual information between  and Y :
[Equation]
The above objective considers taking arg max over not just the discrete set of intervention targets 158 j 2 V , but also over the uncountable set of intervention values v ⇢ X j .
While the existing works 159 in BOECD consider only the design of intervention targets to limit the complexity [48, 29, 1], ourapproach tackles both the problems.
We ﬁrst outline the methodology for a single design and in
Section 3.2 demonstrate how to extend this single design to a batch setting.
3.1 Single Design
To maximize the objective in Equation 3, we need to (1) estimate MI for candidate interventionsand (2) maximize the estimated MI by optimizing over the domain of intervention value for everycandidate interventional target.
Estimating the MI.
As mutual information is intractable, there are various ways to estimateit depending on whether we can sample from the posterior and whether the likelihood can beevaluated [11, 37, 17].
Since the models we consider allow both posterior sampling and likelihoodevaluation, it sufﬁces to obtain an estimator which requires only likelihood evaluation and Monte
Carlo approximations of the expectations.
To do so, we derive an estimator similar to Bayesian Active
Learning by Disagreement (BALD) [17], which considers MI as a difference of conditional entropiesover the outcomes Y :
[Equation]
BO Iterations
N od es
[Figure]
o r
[Figure]
where H( · ) is the entropy.
See Appendix B.1 for the derivation.
A Monte Carlo estimator of the aboveequation can be used as an approximation (Appendix B.2).
Equation (4) has an intuitive interpretation.
It assigns high mutual information to interventions that the model disagrees the most regarding theoutcome.
We denote the MI for a single design as I ( { ( j, v ) } ) :
=
I( Y ;  | { ( j, v ) } , D ) .
Selecting the Intervention Value.
As shown in (3), maximizing the objective is achieved not onlyby selecting the intervention target but also by setting the appropriate intervention value.
Althoughoptimizing the intervention target is tractable (discrete and ﬁnite number of nodes to select from),selecting the value to intervene is usually intractable since they are continuous.
For any given targetnode j , MI is a nonlinear function over v 2 X j
(See Fig 2) and hence solving with gradient ascenttechniques only yields a local maximum.
Given that MI is expensive to evaluate, we treat MI for agiven target node j as a black-box function and obtain its maximum using Bayesian Optimization(BO) [23, 52, 28].
BO seeks to ﬁnd the maximum of this function max v 2 X j I ( { ( j, v ) } ) over the 185 entire set
X j with as few evaluations as possible.
See appendix E for details.
BO typically proceeds by placing a Gaussian Process (GP) [38] prior on the function I ( { j, ·} ) andobtain the posterior over this function with the queried points v ⇤
= { v (1) ⇤ , . . .
, v ( T ) ⇤ } .
Let thevalue of the mutual information queried at each optimization step t be ˆU tj
= I ( { ( j, v ( t ) ⇤ } ) .
Theposterior predictive of a point v ( t +1) can be obtained in closed form as a Gaussian with meanµ ( t +1)
j ( v ) and variance  ( t +1)
j ( v ) .
Subscript j signiﬁes the fact that we maintain different µand  per intervention target and superscript t represents the BO step.
Querying proceeds byhaving an acquisition function deﬁned on this posterior, which suggests the next point to query.
For BO, we use an acquisition function called as the Upper Conﬁdence Bound (UCB) [45] whichsuggests the next point to query by trading-off exploration and exploitation with a hyperparameter t j : v ( t +1) ⇤ j
[Equation]
We run GP-UCB independently on every candidate intervention target j = { 1 , . . .
, d } by queryingpoints within a ﬁxed domain [  k, k ] ⇢ R .
Note that the domain can be chosen based on theapplication, for example, if we must constrain dosage levels within a ﬁxed range.
Each GP isone-dimensional in our setup; hence a few evaluations of UCB are sufﬁcient to get a good valuemaxima candidate.
Further, GP-UCB for each candidate target is parallelizable, making it efﬁcient.
We ﬁnally select the design with the highest MI across the candidate intervention targets.
3.2 Batch Design
In many applications, it is desirable to select the most informative set of interventions instead ofa single intervention at a time.
Take, for example, a biologist entering a wet lab with a script ofexperiments to execute.
Batching experiments removes the bottleneck of waiting for an experimentto ﬁnish and get analyzed until executing the next one.
Given a budget per batch B whichdenotes the number of experiments in a batch, the problem of selecting the batch then becomesarg max ⌅
I( Y ;  | ⌅ , D ) , such that cardinality ( ⌅ ) =
B , where ⌅ is a set of interventions
Input : E environment, N initial observational samples,
B batch Size, d number of nodes
1 ⌅ ? 2 for n = 1 . . .
B do for j = 1 . . .
d do
GP
V j arg max v I ( ⌅ [ { ( j,v ) } )
U j
I ( ⌅ [ { ( j,V j ) } )
j ⇤ arg max j U j v ⇤
V j ⇤ ⌅ ⌅ [ { ( j ⇤ , v ⇤ ) } 9 return ⌅
Input : E environment, N initial observational samples,
B batch Size, d number of nodes, ⇣ softmax temperature 1 for j = 1 . . .
d do
Initialize µ 0 j and  0 j for t = 1 . . .
T do
V tj arg max v µ t
1 j ( v ) + p  t  t  1 j ( v ) ˆ U tj
I ( { ( j, V tj ) } )
Update the GP to obtain µ tj and  tj 7 { ( t i , j i ) } i 2 { 1 ,..., B}
B samples without replacement / exp ( ˆ U tj / ⇣ ) 8 ⌅ { ( j i , V t i j i ) } i 2 { 1 ,..., B} 9 return ⌅
S
B i =
( j i , v i ) and Y denotes the random variable for the outcomes of the interventions of the batch.
We denote the MI for a batch design as I ( ⌅ ) :
=
I( Y ;  | ⌅ , D ) .
Greedy Algorithm.
Computing the optimal solution I ( ⌅ ⇤ ) is computationally infeasible.
However,as the conditional mutual information is submodular and non-decreasing (see Appendix B.4 for proof),we can derive a simple greedy algorithm (Algorithm 1) that can achieve at least a (1  1 /e ) ⇡ 0 .
64approximation of the optimal solution [22, 30].
We denote this strategy as Greedy-CBED .
Soft Top-K.
Although the greedy algorithm is tractable, it requires O ( B d ) instances of GP-UCB.
Kirsch et al.
[20] show that a soft top-k selection strategy performs similarly to the greedy algorithm,reducing the computation requirements to O ( d ) runs of GP-UCB.
To achieve this, we construct aﬁnite set of candidate intervention target-value pairs by keeping all the T evaluations of GP-UCBfor each node j = { 1 , . . .
, d } .
Therefore, for d nodes, we our candidate set is comprised of d ⇥ Texperiments.
We score each experiment in this candidate set using the MI estimate.
We then samplewithout replacement B times proportionally to the softmax of the MI scores (Algorithm 2).
We denotethis strategy as Soft-CBED .
3.3 Comparison with existing active causal discovery methods
We outline how our approach compares with two main existing active causal discovery methods.
ABCD [1].
The estimator of MI used in ABCD is based on weighted importance sampling.
However, for the speciﬁc choice of the importance sampling weights used in ABCD, their MI estimatorends up with the same approximation as in our method (see Appendix B.5).
Nevertheless, ABCDdoes not select intervention values but suboptimally sets them to a ﬁxed value.
In addition, our proposed Soft-CBED is a faster and more efﬁcient batch strategy, especially when values also have to beacquired.
From this perspective, our approach is an extension of ABCD with nonlinear assumptions,value acquisition, and a soft top-k batching strategy.
AIT [41].
AIT is an F-score-based intervention target acquisition strategy.
Although it is not a
BOECD method, we prove here that it can be viewed as a Monte Carlo estimate of the approximationto MI when the outcomes Y are Gaussian.
Nevertheless, AIT does not select intervention values like
ABCD and does not have a batch strategy.
Theorem 3.1.
Let Y be a Gaussian random variable.
Then the discrepancy score of Scherreret al.[41] is a Monte Carlo estimate of an approximation to mutual information (Eq.
See
Appendix B.6 for proof.
Related Work
Early efforts of using Bayesian Optimal Experimental Design for Causal Discovery (BOECD) canbe found in the works of [29] and [48].
However, these approaches deal with simple settings likelimiting the graphs to topologically ordered structures, intervening sequentially, linear models, anddiscrete variables.
In [8] and [31], BOECD was applied for learning biological networks structure.
BOECD was alsoexplored in [14] under the assumption that undirected edges of the graph always forms a tree.
Morerecently, ABCD framework [1] extended the work of [29] and [48] in the setting where interventionscan be applied in batches with continuous variables.
To achieve this, they (approximately) solvethe submodular problem of maximizing the batched mutual information between interventions(experiments), outcomes, and observational data, given a DAG.
DAG hypotheses are sampled using DAG-bootstrap [12].
Our work differs from ABCD in a few ways: we work with both linear andnonlinear SCMs by using state-of-the-art posterior models over DAGs [26], we apply BO to selectthe value to intervene with, but we also prepare the batch using soft BALD [20] which is signiﬁcantlyfaster than the greedy approximation of ABCD method.
In [49] the authors proposed the use of Gaussian Processes to model the posterior over DAGs andthen use BO to identify the value to intervene with, however, this method was not shown to bescalable for larger than bivariate graphs since they rely on multi-dimensional Gaussian Processes formodeling the conditional distributions.
A new body of work has emerged in the ﬁeld of differentiable causal discovery, where the problem ofﬁnding the structure, usually from observational data, is solved with gradient ascent and functional approximators, like neural networks [51, 18, 6, 5].
In recent works [9, 26, 2], the authors proposed a variational approximation of the posterior over the DAGs which allowed for modeling a distribution ratherthan a point estimate of the DAG that best explains the observational data D .
Such work can be used toreplace DAG-bootstrap [12], allowing for the modeling of posterior distributions with greater support.
Besides the BOECD-based approaches, a few active causal learning works have been proposed [16,13, 41, 43, 44, 21].
Active ICP [13] uses ICP [35] for causal learning while using an active policy toselect the target, however, this work is not applicable in the setting where the full graph needs to berecovered.
In [50], the authors propose an active learning method to the problem of identifying theinterventions that push a dynamical causal network towards a desired state.
A few approaches tacklethe problem of actively acquiring interventional data to orient edges of a skeletal graph [43, 44, 21].
Closer to our proposal belongs AIT [41], which uses a neural network-based posterior model overthe graphs but evaluates the F-score to select the interventions.
Experiments
We evaluate the performance of our method on synthetic and real-world causal experimental designproblems and a range of baselines.
We aim to investigate the following aspects empirically: (1)competitiveness of the overall proposed strategies of Greedy-CBED and Soft-CBED at scale ( 50nodes) on synthetic datasets; (2) performance of the value acquisition strategy based on GP-UCB;and (3) performance of the proposed approach on a real-world inspired dataset.
5.1 Acquisition Functions
Random :
Random baseline acquires interventional targets at random.
AIT / soft AIT : active intervention targeting (AIT)
[41] uses an f-score based acquisition strategy to select the interventiontargets.
See appendix B.6 for more details.
Since the original proposed approach does not consider abatch setting, we introduce a variant that augments AIT with the proposed soft batching, as describedin section 3.2.
CBED / GreedyCBED / SoftCBED :
These are the Monte Carlo estimates of MI,as described in section 3.
CBED selects a single intervention (target and value) that maximizes the
MI and this intervention is applied for the whole batch.
In Greedy-CBED , the batch is built up in agreedy fashion selecting the target, value pairs one at a time (Algorithm 1).
Soft-CBED is sampling(target, value) pairs proportionally to the MI scores to select a batch, as described in section 3.2 and
Algorithm 2.
[Figure]
5.2 Value Selection Strategies
Fixed : This value selection strategy assumes setting the value of the intervention to a ﬁxed value.
Inthe experiments, we ﬁxed the value to 0 .
Sample-Dist :
This value selection strategy samples fromthe support of the observational data.
GP-UCB : This strategy uses the proposed GP-UCB Bayesianoptimization strategy to select the value that maximizes MI.
5.3 Tasks
Synthetic Graphs.
In this setting, we generate Erd˝os–Rényi [10] (ER) and Scale-Free (SF)graphs [3] of size 20 and 50.
For linear SCMs, we sample the edge weights  uniformly at random.
For the nonlinear SCM, we parameterize each variable to be a Gaussian whose mean is a nonlinearfunction of its parents.
We model the nonlinear function with a neural network.
In all settings, we setnoise variance  2 = 0 .
For both types of graphs, we set the expected number of edges per vertex to1.
We provide more details about the experiments in appendix D.1.
Single-Cell Protein-Signalling Network.
The DREAM family of benchmarks [15] are designedto evaluate causal discovery algorithms of the regulatory networks of a single cell.
A set of ODEsand SDEs generates the dataset, simulating the reverse-engineered networks of single cells.
We use
GeneNetWeaver [40] to simulate the steady-state wind-type expression and single-gene knockouts .
Refer to appendix D.2 for the exact settings.
Results
[Table] [Table] For each of the acquisition objectives and datasets, we presentthe mean and standard error of the expected structural hammingdistance E -SHD , expected structural interventional distance ESID [33], area under the receiver operating characteristic curve AUROC and area under the precision-recall curve AUPRC .
In Appendix H we describe the metrics in detail.
We evaluate thesemetrics as a function of the number of acquired interventionalsamples (or experiments), which helps quantitatively comparedifferent acquisition strategies.
Apart from E -SID , we relegateresults with other metrics to the appendix I.
[Figure]
On the synthetic graphs (Figure 3(a)), we can see that for ER and SF graphs with 50 D variablesand nonlinear functional relationships, the proposed approach based on soft top-k to select a batchwith GP-UCB outperforms all the baselines in terms of the E -SID metric.
On the other hand, AITalone does not converge to the ground truth graph fast even after combining with the proposed valueacquisition, but when further augmented with the proposed soft strategy, the soft AIT recovers theground truth causal graph upto 4 times faster and performs competitively to Soft-CBED .
We observesimilar performance across other metrics as well.
In addition, we found this trend to hold for 20 Dvariables and linear models.
Full results are presented in the appendix I.
Next, we examine the importance of having a value selection strategy for active causal discovery.
Weuse the MI estimator in Equation 4; moreover, we test the proposed GP-UCB with two heuristics - theﬁxed value strategy and sampling values from the support.
As we can see in Figure 3(b), selectingthe value using GP-UCB clearly beneﬁts the causal discovery process.
We expect this ﬁnding as themutual information is not constant with respect to the intervened value.
To make this point clear, wedemonstrate in the appendix G the inﬂuence of the value in a simple two variables graph.
In addition,we note that naively sampling from the support of the observed dataset performs worse than ﬁxingthe value to 0.
We hypothesize that this is due to lower epistemic uncertainty in the high densityregions of the support, hinting that these regions might be less informative.
In order to further understand how the soft batch strategy compares with other batch selection strategies, we compare the results of Soft-CBED with Greedy-CBED and CBED .
We observe (Figure 3(c))that
Greedy-CBED and Soft-CBED give very similar results overall.
While Greedy-CBED is optimalunder certain conditions [19], Soft-CBED remains competitive and has the advantage that the batchcan be selected in a one-shot manner.
This is also evident from the runtime performance of both thesebatching strategies in Table 1.
Both these batch selection strategies perform signiﬁcantly better thanselecting one intervention target/value pair, and executing them B times ( CBED ).
Finally, on the DREAM task, we see that our method outperforms soft AIT and random baselineson the E -SID metric (see Figure 4).
In these experiments, since the intervention is emulating thegene knockout setting, we only use the ﬁxed value strategy, with a value of 0 . 0 .
Although randombaseline still remains a competitive choice, in certain settings, Soft-CBED objective is signiﬁcantlybetter ( Ecoli1 , Ecoli2 datasets).
Summary and Conclusions
This paper studies the problem of efﬁciently selecting the Bayes optimal experiments to discovercausal models.
Our proposed framework simultaneously answers the questions of where and how tointervene in a batched setting.
We present a Bayesian optimization strategy to acquire interventionaltargets and values.
Further, we propose two different batching strategies: one based on greedyselection and the other based on soft top-k selection.
The proposed methodology for selectingintervention target-value pairs in a batched setting provides superior performance over the state-ofthe-art for causal models up to 50 D variables.
We validate this using synthetic datasets and usingreal-world inspired datasets of single-cell regulatory networks, showing the potential impact on areaslike biology and other experimental sciences.