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 studies the weak topology on three spaces of probability measures, each corresponding to a level of a causal hierarchy introduced by Pearl (Pearl and Mackenzie, 2018).  The three levels, in order of increasing expressivity, are described as observational, interventional and counterfactual; given a set $\bf{V}$ of variables (known as "endogenous" variables; these variables can influence each other and may also be influenced by a set of "exogenous" variables), the observational distribution is a joint probability distribution over the values of $\bf{V}$ fixed according to some given structural causal model, an interventional distribution on $\bf{V}$ is a distribution where the values of some finite subset of $\bf{V}$ are fixed and a counterfactual distribution is one where various combinations of interventions with respect to different finite subsets of $\bf{V}$ are permitted.  The main results are: (1) a topological characterization of the set of hypotheses in the second level of the causal hierarchy that are "experimentally verifiable", which means that there is a sequence of tests that almost surely converge in the limit on the true hypothesis while a type I error is incurred up to a fixed bound at each stage (the relevant distribution belonging to the second level of the hierarchy); (2) the set C_2 of all points in the second level of the causal hierarchy that have a unique inverse (with respect to a projection from the third level) forms a meager set, where the underlying topology is the weak topology; a similar result applies to the preimage of C_2 under projection from the third level of the hierarchy.