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 investigates how interpretation arises within a DAG structure learned from data. The authors proposed Structural Causal Interpretation (SCI) which provides an explanation (interpretation) given data and a why question based on an assumed structural causal model. Under this definition of “interpretation” or “interpretability”, neural-network induced DAG models are considered interpretable. The authors demonstrated the usefulness of SCI.