Dependency Structure Discovery from Interventions

1 INTRODUCTION . Structure learning concerns itself with the recovery of the graph structure of Bayesian networks ( BNs ) from data samples . A natural application of Bayesian networks is to describe cause-effect relationships between variables . In that context , one may speak of causal structure learning . Causal structure learning is challenging because purely observational data may be satisfactorily explained by multiple Bayesian networks ( a Markov equivalence class ) , but only one is the most robust to distributional shifts : The one with the correct graph . A more powerful tool than BNs is thus needed to model causal relationships . Structural Causal Models ( SCMs ) are that tool . An SCM over a set of random variables is a collection of assignments to these variables and a directed acyclic graph of dependencies between them ( Peters et al. , 2017 , §6.2 ) . Each assignment is a function of only the direct causes of a variable , plus an independent noise source . An SCM entails precisely one ( observational ) data distribution . Interventions on an SCM ’ s assignments , such as setting a random variable to a fixed value ( a hard intervention ) , entail new interventional data distributions ( Peters et al. , 2017 , §6.3 ) . SCMs can be used to answer higher-order questions of cause-and-effect , up the ladder of causation ( Pearl & Mackenzie , 2018 ) . Causal structure learning using SCMs has been attempted in several disciplines including biology ( Sachs et al. , 2005 ; Hill et al. , 2016 ) , weather forecasting ( Abramson et al. , 1996 ) and medicine ( Lauritzen & Spiegelhalter , 1988 ; Korb & Nicholson , 2010 ) . Causal structure is most frequently learned from data drawn from observational distributions . Structure learning methods generally can not do more than identify the causal graph up to a Markov equivalence class ( Spirtes et al. , 2000 ) . In order to fully identify the true causal graph , a method must either make restrictive assumptions about the underlying data-generating process , such as linear but non-Gaussian data ( Shimizu et al. , 2006 ) , or must access enough data from outside the observational distribution ( i.e. , from interventions ) . Under certain assumptions about the number , diversity , and nature of the interventions , the true underlying causal graph is always identifiable , given that the method knows the intervention performed ( Heckerman et al. , 1995 ) . In much of the prior work on causal model induction it is assumed that there is an experimenter and this experimenter performs interventions . However , in the real world , interventions can also be performed by other agents , which could lead to unknown interventions ( interventions with unknown target variables ) . A few works have attempted to learn structures from unknown-intervention data ( Eaton & Murphy , 2007a ; Squires et al. , 2020 ; Huang et al. , 2020 ) . A notable such work , ( Mooij et al. , 2016 ) , has been extended in ( Kocaoglu et al. , 2019 ; Jaber et al. , 2020 ) . Although there is no theoretical guarantee that the true causal graph can be identified in that setting , evidence so far points to that still being the case . Another common setting is when the graph structure is partially provided , but must be completed . An example is protein structure learning in biology , where we may have definitive knowledge of some causal edges in the protein-protein interactome , but the remaining causal edges must be discovered . We will call this setting “ partial graph completion ” . This is an easier task compared to learning the entire graph , since it limits the number of edges that have to be learned . Recently , a flurry of work on structure learning using continuous optimization methods has appeared ( Zheng et al. , 2018 ; Yu et al. , 2019 ) . These methods operate on observational data and are competitive with other methods . Because of the theoretical limitations on identification from purely observational data cited above , it would be interesting to extend these methods to interventional data . However , it is not straightforward to apply continuous optimization methods to structure learning from interventional data . Our key contributions are to answer the following questions experimentally : 1 . Can the proposed model recover true causal structure ? Yes , see Figure §4 . 2 . How does the proposed model compare against state of the art causal methods on real-world datasets ? Favourably ; see §5.4 and Table §1 . 3 . Does a proposed model generalize well to unseen interventions ? Yes , see §5.5 . 4 . How does the proposed model perform on partial graph recovery ? It scales to∼ 50 variables while the other baselines can ’ t . see §5.7 . 2 PRELIMINARIES . Causal modeling . A Structural Causal Model ( SCM ) ( Peters et al. , 2017 ) over a finite number M of random variables Xi is a set of structural assignments Xi : = fi ( Xpa ( i , C ) , Ni ) , ∀i ∈ { 0 , . . . , M − 1 } ( 1 ) Identifiability . In a purely-observational setting , it is known that causal graphs can be distinguished only up to a Markov equivalence class . In order to identify the true causal graph structure , interventional data is needed ( Eberhardt et al. , 2012 ) . Interventions . There are several types of common interventions which may be available ( Eaton & Murphy , 2007b ) . These are : No intervention : only observational data is obtained from the ground truth model . Hard/perfect : the value of a single or several variables is fixed and then ancestral sampling is performed on the other variables . Soft/imperfect : the conditional distribution of the variable on which the intervention is performed is changed . Uncertain : the learner is not sure of which variable exactly the intervention affected directly . Here we make use of soft intervention because they include hard intervention as a limiting case and hence are more general . Structure discovery using continuous optimization . Structure discovery is a super-exponential search problem that searches though all possible directed acyclic graphs ( DAGs ) . Previous continuousoptimization structure learning works ( Zheng et al. , 2018 ; Yu et al. , 2019 ; Lachapelle et al. , 2019 ) mitigate the problem of searching in the super-exponential set of graph structures by considering the degree to which a hypothesis graph violates “ DAG-ness ” as an additional penalty to be optimized . If there are M such variables , the strategy of considering all the possible structural graphs as separate hypotheses is not feasible because it would require maintaining O ( 2M 2 ) models of the data.2 3 RELATED WORK . The recovery of the underlying structural causal graph from observational and interventional data is a fundamental problem ( Pearl , 1995 ; 2009 ; Spirtes et al. , 2000 ) . Different approaches have been studied : score-based , constraint-based , asymmetry-based and continuous optimization methods . Score-based methods search through the space of all possible directed acyclic graphs ( DAGs ) representing the causal structure based on some form of scoring function for network structures ( Heckerman et al. , 1995 ; Chickering , 2002 ; Tsamardinos et al. , 2006 ; Hauser & Bühlmann , 2012 ; Goudet et al. , 2017 ; Cooper & Yoo , 1999 ; Zhu & Chen , 2019 ) . Constraint-based methods ( Spirtes et al. , 2000 ; Sun et al. , 2007 ; Zhang et al. , 2012 ; Monti et al. , 2019 ; Zhu & Chen , 2019 ) infer the DAG by analyzing conditional independences in the data . Eaton & Murphy ( 2007c ) use dynamic programming techniques to accelerate Markov Chain Monte Carlo ( MCMC ) sampling in a Bayesian approach to structure learning for discrete variable DAGs . Peters et al . ( 2016 ) ; Ghassami et al . ( 2017 ) ; Rojas-Carulla et al . ( 2018 ) exploit invariance across environments to infer causal structure , which faces difficulty scaling due to the iteration over the super-exponential set of possible graphs . Recently , ( Zheng et al. , 2018 ; Yu et al. , 2019 ; Lachapelle et al. , 2019 ) framed the structure search as a continuous optimization problem , however , the methods only uses observational data and are non-trivial to extend to interventional data . In our paper , we present a method that uses continuous optimization methods that works on both observational and interventional data . For interventional data , it is often assumed that the models have access to full intervention information , which is rare in the real world . Rothenhäusler et al . ( 2015 ) have investigated the case of additive shift interventions , while Eaton & Murphy ( 2007b ) have examined the situation where the targets of experimental interventions are imperfect or uncertain . This is different from our setting where the intervention is unknown to start with and is assumed to arise from other agents and the environment . Learning based methods have been proposed ( Guyon , 2013 ; 2014 ; Lopez-Paz et al. , 2015 ) and there also exist recent approaches using the generalization ability of neural networks to learn causal signals from purely observational data ( Kalainathan et al. , 2018 ; Goudet et al. , 2018 ) . Neural network methods equipped with learned masks , such as ( Ivanov et al. , 2018 ; Li et al. , 2019 ; Yoon et al. , 2018 ; Douglas et al. , 2017 ) , exist in the literature , but only a few ( Kalainathan et al. , 2018 ) have been adapted to causal inference . This last work is , however , tailored for causal inference on continuous variables and from observations only . Adapting it to a discrete-variable setting is made difficult by its use of a Generative Adversarial Network ( GAN ) Goodfellow et al . ( 2014 ) framework . 4 STRUCTURE DISCOVERY FROM INTERVENTIONS METHOD . Scope of Applicability and Objective . The proposed method , like any structure learning algorithm , assumes the availability of a data-generating process based on ancestral sampling of a ground-truth SCM of M variables , which can be queried for samples . The SCM supports applying and retracting known or unknown interventions . The method can support infinite- or finite-data as well as infiniteor finite-intervention regimes . The objective is , then , to learn the SCM ’ s structure from the insights that each intervention gives about cause-effect relationships between variables in the SCM . 4.1 PROBLEM SETTING AND ASSUMPTIONS . In this paper , we restrict the problem setting to specific , but still broad classes of SCMs and interventions . In particular , we assume that : Data is discrete-valued . The SCM ’ s random variables are all categorical . Causal sufficiency . For every data sample , the value of all variables are available ; There are no latent confounders . Interventions are localized . They affect only a single variable ( but which one may not be known ) . Interventions are soft . An intervention does not necessarily pin its target random variable to a fixed value ( though it may , as a special case ) . It changes the relationship of a variable with its parents . Interventions do not stack . Before a new intervention is made , the previous one is fully retracted . This stops the SCM from wandering away from its initial , observational configuration after a long series of interventions . No control over interventions . The structure learning algorithm has control neither of the target , nor the nature of the next intervention on the SCM . For a detailed description of the interventions , refer to §A.2 .

Summary: This paper aims to extend the continuous optimization approach to causal discovery to handle interventional data as well as observational data. It describes a method for learning the causal structure over a set of categorical variables and reports strong empirical performance. However, no theoretical guarantee or analysis is provided, which is a significant weakness in my view. It also makes no comment on or comparison to a paper that has essentially the same goal, https://arxiv.org/pdf/2007.01754.pdf. The latter paper seems to me more principled and convincing.