A Topological Perspective on Causal Inference

1 Introduction and Motivation . In the background of any investigation into learning algorithms are no-free-lunch phenomena : roughly , the observation that assumption-free statistical learning is infeasible in general ( see , e.g. , [ 33 , Ch . 5 ] for a formal statement ) . Common wisdom is that learning algorithms and architectures must adequately reflect non-trivial features of the data-generating distribution to gain inductive purchase . For many purposes we need to move beyond passive observation , focusing instead on what would happen were we to act upon a given system . Even further , we sometimes desire to explain the behavior of a system , raising questions about what would have occurred had some aspects of a situation been different . Such questions depend not just on the data distribution ; they depend on deeper features of underlying data-generating processes or mechanisms . It is thus generally acknowledged that stronger assumptions are required if we want to draw causal conclusions from data [ 35 , 28 , 20 , 30 , 32 ] . Whether implicit or explicit , any approach to causal inference involves a space of candidate causal models , viz . data-generating processes . Indeed , a blunt way of incorporating inductive bias is simply to omit some class of possible causal hypotheses from consideration . Many ( im ) possibility results in the literature can accordingly be understood as pertaining to all models within a class . For instance , if we can restrict attention to Markovian models that satisfy faithfulness , then we can always identify the structure of a model from experimental data ( e.g. , [ 11 , 35 ] ) . If we can restrict attention to Markovian ( continuous ) models with linear functions and non-Gaussian noise , then every model can be learned even from purely observational data [ 34 ] . As a negative example , in the larger class of ( not necessarily Markovian ) models , no model can ever be determined from observational data alone [ 35 , 2 ] . At the same time , in many settings it is sensible to aim for results with “ nearly universal ” force . It is natural to ask , e.g. , within the class of all Markovian models , how “ typical ” are those in which the faithfulness condition is violated ? This might tell us , for instance , how typically we could expect failure of a method that depended on these assumptions . A well-known result shows that , fixing 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . any particular causal dependence graph , such violations have measure zero for any smooth ( e.g. , Lebesgue ) measure on the parameter space of distributions consistent with that graph [ 24 ] . In fact , the standard notion of statistical consistency itself , which underlies many possibility results in causal inference , requires omission of some purportedly “ negligible ” set of possible data streams [ 9 , 35 ] . There are two standard mathematical approaches to making concepts like “ typical ” and “ negligible ” rigorous : measure-theoretic and topological . While the two approaches often agree , they capture slightly different intuitions [ 25 ] . One virtue of the measure-theoretic approach is its natural probabilistic interpretation : intuitively , we are exceedingly unlikely to hit upon a set with measure zero . At the same time , the measure-theoretic approach is sometimes criticized in statistical settings for its alleged dependence on a measure , and this has been argued to favor topological approaches ( see , e.g. , [ 3 ] on no-free-lunch theorems ) . The latter of course in turn demands an appropriate topology . In the present work we show how to define a sequence of meaningful topologies on the space of causal models , each corresponding to a progressively coarser level of the so called causal hierarchy ( [ 29 , 2 ] ; see Fig . 1 for an abbreviated pictorial summary ) . We aim to demonstrate that topologizing causal models in this way helps clarify the scope and limits of causal inference under different assumptions , as well as the potential empirical status of those very assumptions , in a highly general setting . Our starting point is a canonical topology on the space of Borel probability distributions called the weak topology . The weak topology is grounded in the fundamental notion of weak convergence of probability distributions [ 4 ] and is thereby closely related to problems of statistical inference ( see , e.g. , [ 8 ] ) . Recent work has sharpened this correspondence , showing that open sets in the weak topology correspond exactly to the statistical hypotheses that can be naturally deemed verifiable [ 14 , 16 ] . We extend the correspondence to higher levels of the causal hierarchy , including the most refined and expansive “ top ” level consisting of all ( well-founded ) causal models . Lower levels and natural subspaces ( e.g. , corresponding to prominent causal assumption classes ) emerge as coarsenings and continuous projections of this largest space . As an illustration of the general approach , we prove a topological version of the causal hierarchy theorem from [ 2 ] . Rather than showing that collapse happens only in a measure zero set as in [ 2 ] , our Theorem 3 show that collapse is topologically meager . Conceptually , this highlights a different ( but complementary ) intuition : not only is collapse exceedingly unlikely in the sense of measure , meagerness implies that collapse could never be statistically verified . Correlatively , this implies that any causal assumption that would generally allow us to infer counterfactual probabilities from experimental ( or “ interventional ” ) probabilities must itself be statistically unverifiable ( Corollary 1 ) . To derive such a result we actually show something slightly stronger ( see Lem . 2 ) : even with respect to the subspace of models consistent with a fixed temporal order on variables , the causal hierarchy theorem holds . Merely knowing the temporal order of the variables is not enough to render collapse of the hierarchy a statistically verifiable proposition . Furthermore , we show that the witness to collapse can be taken as any of the well-known counterfactual “ probabilities of causation ” ( see , e.g. , [ 27 ] ) : probabilities of necessity , sufficiency , necessity and sufficiency , enablement , or disablement . That is , none of these important quantities are fully determined by experimental data except in a meager set . In §2 we give background on causal models , and in §3 we present a model-theoretic characterization of the causal hierarchy as a sequence of spaces . Topology is introduced in §4 , and the main results about collapse appear in §5 . For the technical results , we include proof sketches in the main text to provide the core intuitions , relegating some of the details to an exhaustive technical appendix , which also includes additional supplementary material . 2 Structural Causal Models . A fundamental building block in the theory of causality is the structural causal model [ 26 , 35 , 28 ] or SCM , which formalizes the notion of a data-generating process . In addition to specifying datagenerating distributions , these models also specify the generative mechanisms that produce them . For the purpose of causal inference and learning , SCMs provide a broad , fine-grained hypothesis space . The notions in this section have their usual definition following , e.g. , [ 28 ] , but we have recast them in the standard language of Borel probability spaces so as to handle the case of infinitely many variables rigorously . We start with notation , basic assumptions , and some probability theory . Notation . The signature ( or range ) of a variable V is denoted χV . Where S is a set of variables , let χS =×S∈S χS . Given an indexed family of sets { Sβ } β∈B and elements sβ ∈ Sβ , let ( sβ ) β denote the tuple whose element at index β is sβ , for all β . For B′ ⊂ B write πB′ : ×β∈B Sβ →×β∈B′ Sβ for the projection map sending each ( sβ ) β∈B 7→ ( sβ′ ) β′∈B′ ; abbreviate πβ′ = π { β′ } , where β′ ∈ B . The reader is referred to standard texts [ 21 , 5 ] for elaboration on the concepts used below . Definition 1 ( Topology ) . For discrete spaces ( like χS , for a single categorical variable S ) we use the discrete topology and for product spaces ( like χS for a set of variables S ) we use the product topology . Note that the so-called cylinder sets of the form π−1Y ( { y } ) for finite subsets Y ⊂ S and y ∈ χY form a basis for the product topology on χS . This cylinder set is a subset of χS , and contains exactly those valuations agreeing with the value πY ( y ) specified in y for Y , for every Y ∈ Y . Following standard statistical notation this cylinder is abbreviated as simply y . Definition 2 ( Probability ) . Where ϑ is a topological space write B ( ϑ ) for its Borel σ-algebra of measurable subsets . Let P ( ϑ ) be the set of probability measures on B ( ϑ ) . Specifically , elements of P ( ϑ ) are functions µ : B ( ϑ ) → [ 0 , 1 ] assigning a probability to each measurable set such that µ ( ϑ ) = 1 and µ ( ⋃∞ i=1 ( Si ) ) = ∑∞ i=1 µ ( Si ) for each sequence S1 , S2 , . . . of pairwise disjoint sets from B ( ϑ ) . A map f : ϑ1 → ϑ2 is said to be measurable if f−1 ( S2 ) ∈ B ( ϑ1 ) for every S2 ∈ B ( ϑ2 ) . Fact 1 ( Lemma 1.9.4 [ 5 ] ) . A Borel probability measure is determined by its values on a basis . 2.1 SCMs , Observational Distributions . Let V be a set of endogenous variables . We assume for simplicity every variable V ∈ V is dichotomous with χV = { 0 , 1 } , although the results here generalize to any larger countable range . Influences among endogenous variables are the main phenomena our formalism aims to capture . A well-founded1 direct influence relation→ on V encapsulates the notion of one endogenous variable possibly influencing another . For each V ∈ V , we call { V ′ ∈ V : V ′ → V } = Pa ( V ) the parents of V. We assume every set Pa ( V ) is finite ; this condition is called local finiteness . These two assumptions ( well-foundedness and local finiteness ) generalize the common recursiveness assumption to the infinitary setting , and have an alternative characterization in terms of “ temporal ” orderings : Fact 2 . Say that a total order ≺ on V is ω-like if every node has finitely many predecessors : for each V ∈ V , the set { V ′ : V ′ ≺ V } is finite . Then the influence relation→ is extendible to an ω-like order iff→ is well-founded and locally finite . In addition to endogenous variables , causal models have exogenous variables U . Each endogenous V depends on a subset U ( V ) ⊂ U of “ exogenous parents ” and uncertainty enters via exogenous noise , that is , a distribution from P ( χU ) . A structural function ( or mechanism ) for V ∈ V is a measurable fV : χPa ( V ) × χU ( V ) → χV mapping parental endogenous and exogenous valuations to values . Definition 3 . A structural causal model is a tuple M = 〈U , V , { fV } V ∈V , P 〉 where U is a collection of exogenous variables , V is a collection of endogenous variables , fV is a structural function for each V ∈ V , and P ∈ P ( χU ) is a probability measure on ( the Borel σ-algebra of ) χU . As is well known , recursiveness implies that each u ∈ χU induces a unique v ∈ χV that solves the simultaneous system of structural equations { V = fV } V : Proposition 1 . Any SCMM with well-founded , locally finite parent relation→ induces a unique measurable mM : χU → χV such that fV ( πPa ( V ) ( m M ( u ) ) , πU ( V ) ( u ) ) = πV ( mM ( u ) ) for all u ∈ χU and V ∈ V. Measurability then entails that the exogenous noise P induces a distribution on joint valuations of V , called the observational distribution , which characterizes passive observations of the system . Definition 4 . The observational distribution pM ∈ P ( χV ) is defined on open sets by pM ( y ) = P ( ( mM ) −1 ( y ) ) . Here recall that y represents a cylinder subset ( Definition 1 ) of χV . 1See Appendix A for additional background on orders and relations .

Summary: This paper presents a topological framework to theorize about causal inference, and defines a weak topology on each of three spaces of probability measures, corresponding to the three levels of Pearl's causal hierarchy. In this setup a topological version of the causal hierarchy theorem is derived, to the effect that collapse of the hierarchy is topologically negligible. An interesting learning theoretic interpretation of this result is also given, in the spirit of no-free-lunch theorems.