Structural Causal Interpretation Theorem

Human mental processes allow for qualitative reasoning about causality in terms of mechanistic relations of the variables of interest , which we argue are naturally described by structural causal model ( SCM ) . Since interpretations are being derived from mental models , the same applies for SCM . By defining a metric space on SCM , we provide a theoretical perspective on the comparison of mental models and thereby conclude that interpretations can be used for guiding a learning system towards true causality . To this effect , we present a theoretical analysis from first principles that results in a human-readable interpretation scheme consistent with the provided causality that we name structural causal interpretations ( SCI ) . Going further , we prove that any existing neural induction method ( NIM ) is in fact interpretable . Our first experiment ( E1 ) assesses the quality of such NIM-based SCI . In ( E2 ) we observe evidence for our conjecture on improved sample-efficiency for SCI-based learning . After conducting a small user study , in ( E3 ) we observe superiority in human-based over NIM-based SCI in support of our initial hypothesis . 1 INTRODUCTION . There has been an exponential rise in the use of machine learning , especially deep learning in several real-world applications such as medical image analysis ( Ker et al. , 2017 ) , particle physics ( Bourilkov , 2019 ) , drug discovery ( Chen et al. , 2018 ) and cybersecurity ( Xin et al. , 2018 ) to name a few . While there have been several arguments that claim deep models are interpretable , the practical reality is much to the contrary . The very reason for the extraordinary discriminatory power of deep models ( namely , their depth ) is also the reason for their lack of interpretability . To alleviate this shortcoming , interpretable machine learning ( Chen et al. , 2019 ; Molnar , 2020 ) has gained traction to explain algorithm predictions and thereby increase the trust in these learned models . In their seminal book , Pearl & Mackenzie ( 2018 ) argue that causal reasoning is the most important factor for machines to achieve true human-level intelligence . The same has been pointed out recently by Hofman et al . ( 2021 ) who argue that systems that are efficient in both causality and interpretations are need of the hour . Questions of the form ” What if ? ” and ” Why ? ” have been shown to be used by children to learn and explore their external environment ( Gopnik , 2012 ; Buchsbaum et al. , 2012 ) and are essential for human survival ( Byrne , 2016 ) . This makes understanding and reasoning about causality an inherently important problem . While acknowledging the difficulty of the problem , we push in this direction pragmatically by presenting the first work on causal interpretations—interpreting a ( deep ) causal induction method by grounding it in its causal semantics , which we call structural causal interpretations ( SCI ) . We show how neural causal induction models due to their causal semantics are interpretable , but then go another step , by also suggesting how these interpretations , if available , can be used in the first place to improve on our models understanding , thereby also establishing their importance in the process . The motivation behind this work is to move beyond black-box heat map based methods in explainable artificial intelligence . Although there exist some works such as explanatory interactive learning ( XIL ) ( Teso & Kersting , 2019 ) and Clever-Hans methods ( Lapuschkin et al. , 2019 ; Stammer et al. , 2021 ) that do move beyond such heat maps , we can go a step further and provide structured causal interpretations . XIL methods can fix Clever-Hans like moments but are overly dependent on the expert provided explanations . SCIs provide an elegant way to circumvent the Clever-Hans moments while also avoiding external factors such as expert explanations . Thus , we set the foundation for further research into making neural models , without initial causal semantics , possibly more causal . Overall , we make the following contributions : ( 1 ) Starting from first principles involving mental models , causality and interpretations , we establish SCI theoretically to provide humanunderstandable interpretations from causal models , ( 2 ) we prove that SCI are readily available and that in fact any neural induction model ( NIM ) is SCI-interpretable , ( 3 ) we show empirically for three example members of NIM ( Zheng et al . ( 2018 ) , Goudet et al . ( 2018 ) , Yu et al . ( 2019 ) ) multiple example SCIs and assess their quality relative to the true underlying causality , ( 4 ) we show that existing SCIs can help in improving the learning-based causal induction , ( 5 ) we conduct a small human study to judge the qualitative causal structure for some graphs and show that human-based SCIs are close to the ground truth interpretations . We make our code publicly available.1 2 BACKGROUND AND RELATED WORK . Let us briefly review the background on a recent stride in learning that proposes a continuous formulation of data-driven estimation of ( causal ) graph structures , then the formalism of structural causal models and how estimation of respective dependency terms is conducted , and finally consider established works on the abstract high level ideas of explanations and interpretations . Learning Directed Acyclic Graphs . Induction of inter-variable relationships based on available data lies at the core of most scientific endeavour ( Penn & Povinelli , 2007 ) . The sub-class of relation structures known as Directed Acyclic Graphs ( DAG ) is being focussed here due to its representational role in causality ( Pearl , 2009 ; Peters et al. , 2017 ) . Unfortunately , due to the combinatoric nature of the problem setting , learning DAGs from data is recognized to be an NP-hard problem ( Chickering et al. , 2004 ) . In their seminal work , Zheng et al . ( 2018 ) were able to re-formulate the traditional view into a continuous shape such that any non-convex optimization module can be applied to tackle the graph estimation problem . The authors propose the general formulation , min W∈Rd×d f ( W ) subject to h ( W ) = 0 , ( 1 ) where f is a data-based score , e.g . in Zheng et al . ( 2018 ) a regularized least-squares loss is applied assuming a sparse linear SCM i.e. , f ( W ) = ||X −XW ||2F + ||W ||1 , and h is a smooth function with a kernel ( or null space ) that only contains acyclic graphs , h ( W ) = 0 ⇐⇒ W is acyclic . For the acyclicity constraint , different variations of the same continuous counting mechanism have been proposed , e.g. , Zheng et al . ( 2020 ) proposed h ( W ) = tr ( eW ◦W ) −d while Yu et al . ( 2019 ) proposed h ( W ) = tr [ ( I +W ◦W ) m ] −m , unfortunately , both suffer from cubic runtime-scalability in the number of graph nodes O ( d3 ) . While the aforementioned works have focussed on data originating from ( non-linear transformation ) of linear SCM , there exists yet another sub-class of DAG-learning methodologies that focuses on more general causal inference . Ke et al . ( 2019 ) made use of data from the first two levels of Pearl ’ s Causal Hierarchy ( PCH ) ( Pearl , 2009 ; Bareinboim et al. , 2020 ) , namely observational and interventional , to update their graph estimate W = G ( C ) while using masked neural networks to mimic the structural equations fi = MLPθ ( pa ( Xi ) ) in order to maximize the likelihood of the data under given parameterization . Brouillard et al . ( 2020 ) follows the same idea of leveraging causal information , e.g . interventional data , for overcoming identifiability issues while staying close to the continuous optimization formalism introduced in 1 . Causal Models and Effect Estimation . Following Peters et al . ( 2017 ) , a Structural Causal Model ( SCM ) is defined as C : = ( S , PN ) where PN is a product distribution over noise variables and S is defined to be a set of d structural equations Xi : = fi ( pa ( Xi ) , Ni ) , where i = 1 , . . . , d ( 2 ) with pa ( Xi ) representing the parents of Xi in graph G ( C ) . An SCM C induces a DAG G with edges ( i , j ) ∈ V × V meaning i causes j , induces an observational/associational distribution pC , it can be intervened upon using the do-operator and thus generate interventional distributions pC ; do ( ... ) and furthermore given some observations v can also be queried for interventions within a system with fixed noise terms amounting to counterfactual distributions pC|V=v ; do ( ... ) , that is , C =⇒ PCH . To query for samples of a given SCM , the structural equations are being simulated sequentially following the underlying causal structure starting from the independent , exogenous variables . The dependency terms provided by the structural equations fi defines the translation between domains 1 https : //anonymous.4open.science/r/Structural-Causal-Interpretation-Theorem-D5C0 of the cause and the effect i.e. , the causal effect g ( X → Y ) . In most settings , this causal effect , e.g. , of a medical treatment onto the patient ’ s recovery , is the sought quantity of interest . If interventions are admissible , then the average causal ( or treatment ) effect ( ACE/ATE ) within a binary system is defined as a difference in interventional distributions : g ( X → Y ) : = ACE ( X , Y ) = E [ Y | do ( X = 1 ) ] − E [ Y | do ( X = 0 ) ] . The ACE overcomes the fundamental problem of causal inference , which states the restriction that individual-level causal effects can not be estimated as we can not observe counterfactuals , but still suffers from the problem of identifiability i.e. , that we usually don ’ t even have access to interventions and thus need to resort to other causal knowledge ( e.g . structural knowledge on confounders ) . Explanations and Interpretations in Deep Learning . A great body of work within deep learning has provided visual means for explanations of how a neural model came up with its decision i.e. , importance estimates for a model ’ s prediction are being mapped back to the original input space e.g . raw pixels ( Selvaraju et al. , 2017 ; Sundararajan et al . ; Schwab & Karlen , 2019 ; Schulz et al. , 2020 ) . Stammer et al . ( 2021 ) argue that such explanations are insufficient for any task that requires symbolic-level knowledge while comparing the existing state of explanations to children that are only able to point fingers but lack articulation . While agreeing with the aspect that simple heatmap explanations are indeed insufficient for reasoning tasks , we consider the higher order task of understanding the meaning of a matter ( why ) to be akin to what is meant by the word interpretation , while an explanation ( invariant to its instantiation ) refers to identifying the matter of interest ( what ) . Seen from this perspective , explanations are a crucial sub-process within interpretation . Thereby what the authors in ( Stammer et al. , 2021 ) seek is in fact an interpretation i.e. , the articulation by the child which also takes into account the causal dynamics of the scene .

Summary: This paper proposes a new concept named structural causal interpretation (SCI) and the theory supporting SCI and its applications. Several contributions are made by the paper, including: (1) The theory and method for automatic generation of human understandable interpretations (i.e. SCIs) from causal models, including deep causal induction models. (2) The theory and algorithm for using existing SCIs (e.g. obtained from human) to improve the learning of causality by deep causal induction models. (3) Experiments and the small human study for illustrating the theory and methods.