Symmetries and Causality: Causal Effect Identification Beyond IID Data

1 Introduction

• The paper targets causal reasoning for complex real-world and AI problems where conventional do-operator, graphical-model, and potential-outcome formalisms have limited applicability and expressiveness.Its motivation is that causal reasoning may support efficient world modeling, while existing frameworks often assume nearly IID data and answer only intervention queries.Source: §1 · p. 1

1.1 Need for Abstraction

• The proposed abstraction removes IIDness and randomness from model descriptions, enabling unified treatment of data from different environments, experiments, and stochastic processes without adding context variables.Probability kernels replace ambiguous structural-mapping and noise-distribution pairs, while preserving rigorous causal reasoning and accommodating stochastic data.Source: §1.1 · pp. 2–3

1.2 Employed Structure

• Mechanism invariance is elevated to the formal starting point: shared models describe observed and intervened worlds, with invariant kernels supporting known interventions and data-based knowledge extraction.The framework explicitly attaches symmetries and applicability ranges, and in the IID do-interventional case reproduces SCM and potential-outcome results.Source: §1.2 · pp. 3–4

1.3 Contributions and Content

• The primary contribution is formal machinery reconnecting abstract model descriptions to identification strategies, organized into model computation, knowledge extraction from observations, and causal-query formulation.The paper provides algorithmic descriptions but does not focus on implementation, excludes selection bias and correlated missingness, and leaves broader completeness questions open.Source: §1.3 · pp. 4–5

1.4 Related Literature

• Relative to prior work on transportability, causal discovery, invariance, and abstract causality, the paper focuses on identifying complex non-IID causal queries, while excluding structure discovery and discussing links to mediation and missing-data problems [5] [33].It recovers analogues of c-components, c-trees, c-forests, and hedges, and its flexible queries capture some natural direct effects; structure discovery remains outside scope.Source: §1.4 · pp. 5–6

2.1 Preliminaries

• The preliminaries assume standard Borel measurable spaces, define probability kernels, and represent symmetries through group actions, invariance, and equivariance.These assumptions prioritize causal properties over measure-theoretic technicalities.Source: §2.1 · p. 6

2.2 Model

• A model partitions a countable index set into regions whose mechanisms are probability kernels with freely transitive symmetries, distinguishing unknown observational from known interventional mechanisms.The framework represents IID and time-series models, and unifies experimental or multicontext settings through a combined index set.Source: §2.2 · pp. 6–7

2.3 Shallow Distribution and Realization

• Under an acyclic index graph and finite past, the model defines a shallow distribution and an observable world that exists, is unique in distribution, and is locally Markov.The shallow distribution records at most one observation per variable, so statistical analysis requires invariant subsets that grow asymptotically.Source: §2.3 · pp. 7–8

2.4 Observedness and Asymptotics

• Observedness is encoded by a deterministic viewport selecting variables across an infinite index sequence, with independent missingness and one observed value per viewport element.A realized dataset combines the observable world with the viewport, while valid transfer remains restricted to observational support.Source: §2.4 · p. 9

3.1 Preliminaries

• Kernel calculus supports ordered products, marginalizations, disintegrations, reordering, and composition, with finite regular computations uniquely defined; completeness of this class is conjectured.These operations generalize conditional-distribution manipulations while respecting causal order, and Conjecture A states that every finite uniquely defined computation on generic kernels is regular.Source: §3.1 · pp. 9–10

3.2 Causality and Sparsity

• Model-aligned acyclic structural graphs encode sparse parent relationships by wiring node-specific kernels to their parents, yielding structured kernels with shared outer arguments and optional hidden inner nodes.The construction preserves parent value-spaces as child-domain factors and defines F-structured kernels when each component kernel belongs to a class F.Source: §3.2 · pp. 10–11

3.3 Graphical Operations

• Graphical simplification, c-component computation, and gluing provide regular functionals for reducing structured kernels, extracting c-components, and reconstructing larger kernels from smaller constituents.The c-component and gluing lemmas establish computability from the original kernel or compatible subgraphs, respectively.Source: §3.3 · p. 12
• Revealing enables non-parametric causal missing-value imputation by changing latent sets within structural c-components, while decomposition and gluing can be ordered and implemented algorithmically.The section conjectures that every regular computation of a structured kernel can ultimately be represented by a finite sequence of graphical operations.Source: §3.3 · p. 12

4 Extraction from Data

• The main text specializes to the single-level case for simpler notation, while the appendix gives results and proofs for the multi-level case.Multi-context systems can also be represented in a single-level system when relations between context variables are irrelevant to the query.Source: §4 · p. 13, §4.6 · p. 18

4.1 Families of Embeddings

• Local graph embeddings injectively map model-aligned structures into the I-graph while including proper parents, preserving proper edges, and aligning inner nodes with model mechanisms.Families collect repeated occurrences of one local structure, using an anchor node, symmetry, and applicability range.Source: §4.1 · p. 13

4.2 Decorated Families

• Decorated families augment embeddings with minimal latent subsets and valid minimal ancestral structures, defining backdoor-freeness when latent nodes are not outer and no ancestral edge starts at a latent node.These structures are generally non-unique, and extraction grows embeddings by absorbing latent parents or backdoor children until the family is backdoor-free.Source: §4.2 · pp. 14–15

4.3 Notions of Identifiability

• Identifiability is defined from valid repeated data-sets: a kernel is identifiable when uniquely computable from finitely many directly identifiable kernels through a regular functional.Unlike IID settings, the shallow distribution may not be learnable because variables in the observable world can be observed only once; direct identifiability formalizes repeated observations instead.Source: §4.3 · pp. 16–17

4.4 Identifiability from Embeddings

• Backdoor-free families of embeddings make μ(G, L) identifiable, because inner nodes form unions of structural c-components when latent-to-outer backdoor paths are absent.The paper presents this as Theorem 1 and relates the result to a variant of Lemma 3.7.Source: §4.4 · p. 17

4.5 Summary: Extracted Knowledge

• The extracted knowledge set comprises all backdoor-free embedding families and their identifiable structured models, together with known intervention mechanisms represented as identifiable single-level models.For each backdoor-free family, the associated structured model is k(ψ, L, A) := (G, L); known elements of Fintervene are identifiable by definition.Source: §4.5 · p. 17

4.6 Extension: Multi-Level Statistics

• The framework supports emergent hierarchy by allowing finite parent sets and arbitrarily many children, so context variables can organize site-specific mechanisms and hidden variables across time-points.The same variable may act as a context for one extraction task and an ordinary variable for another.Source: §4.6 · pp. 17–18

5 Queries and Prediction

• The section frames causal questions as basic queries made graphically accessible through structured queries, with identifiability following from the preceding kernel-extraction and graphical-transformation machinery.The framework is intended to state abstract identification results independently of embeddings or structured kernels and is complete in the IID case under the stated construction.Source: §5 · pp. 18–19, §5.1 · p. 19

5.1 Basic Queries

• Basic queries formalize finite target and conditioning sets with intervention parameters, while query-identifiability means the associated probability kernel is identifiable from the dataset.Interventional queries use an empty conditioning set, whereas nonempty conditioning sets support intermediate results.Source: §5.1 · p. 19

5.2 Structured Queries

• Structured queries embed local graphs with an interest set, induce underlying basic queries, and can be identified when their regular functional decomposes into identifiable components.EDAIdentify extracts backdoor-free families, decomposes the resulting knowledge, and assembles it into the query graph; completeness remains conditional on stated conjectures.Source: §5.2 · pp. 19–20

5.3 Knowledge-Closures and Identification

• Knowledge-closures characterize identification abstractly as recovery of knowledge through a regularity class closed under finite composition.With standard regular functionals, a quantity is identifiable exactly when it belongs to the closure of Kbasic; instrumental-variable deconvolution is included through a broader closure class.Source: §5.3 · pp. 20–21

5.4 Relation to the IID-Case

• The approach matches complete IID identification for unconditional do-interventions and mz-transportability, and reproduces natural direct-effect mediation formulas without counterfactuals.For conditional queries, internal do-intervention structure can be represented as meta-queries; the authors report no regression below the state of the art in standard setups.Source: §5.4 · p. 21

• The formalism uses weak imposed structure to support learning and reasoning about models and identifying causal queries that include conventional questions as special cases.These developments are presented across §§2–5.Source: §6 · p. 22
• The framework rediscovers c-components, c-trees, c-forests, and hedges, reproduces the mediation formula for natural direct effects and mz-transport results, and remains incomplete for several questions.Open areas include selection bias, correlated missingness, counterfactuals, and completeness of the identification strategy.Source: §6 · p. 22
• Learning the formalism’s models from data remains an open direction, with discovering the right structure necessarily preceding methods for learning it.The paper points to §F.3 for this broader learning question.Source: §6 · p. 22
• In the introductory example, Thm. 1 identifies the highlighted “learn” panels, lemma 3.8 glues these partial results with known Yinit, and irrelevant variables are then marginalized.Source: §6 · p. 22

Reading Guide

• Multi-level results are removed from the main text for readability, then introduced in §C and §D as modifications that enable more general query formulations.In the single-level case, marginalization simplifies queries, whereas multi-level examples and applications appear in notation C.11 and §D.5.Source: Reading Guide · p. 26
• The paper separates observable-world randomness from the measure-theoretic model, using §G to connect textbook results with its notation and computational methods.§G also recalls kernels and explains basic notions through structural causal model examples.Source: Reading Guide · p. 26, A Details on Structured Kernels · p. 26

A Details on Structured Kernels

• Large tensor-like products of sparse kernels motivate graphical representations that organize causal sparsity information and support reasoning about structured kernels.The appendix defines a precise graph representation because tensor networks and causal graphical models both encode interdependencies graphically.Source: §A.1 · p. 26

A.1 Structural Graphs

• A structural graph is a finite set of inner and outer nodes with directed proper edges into inner nodes; acyclicity induces a partial order extendable to a generally nonunique causal order.The edge convention excludes self-edges and allows at most one edge per orientation between nodes.Source: §A.1 · pp. 26–27
• Repeated parent use can be represented by diagonal-map soft interventions, but observing only the resulting composition cannot identify predictions requiring the full multi-argument mechanism.If fY is identifiable, composing it with the diagonal intervention remains identifiable; the reverse direction generally loses information.Source: §A.1 · p. 27
• Model alignment attaches probability kernels to inner nodes and bijectively assigns kernel arguments to proper parents, enabling graph-conforming computation in causal order.Structural contraction multiplies aligned node kernels in causal order while leaving selected outer nodes open as arguments.Source: §A.1 · p. 27
• A structured kernel combines a model-aligned acyclic graph with latent-structure information L, producing an F-structured kernel when every inner node kernel belongs to F.The construction depends on the causal order only up to transposition, and structural subgraphs preserve induced edges and inner-node kernels.Source: §A.1 · pp. 27–28

Support and transfer

• Observational data identify conditional kernels only almost everywhere on their support, so transfers such as singular do-interventions may be formally undefined.Two models can share the same observational distribution yet disagree on P(Y|do(X = 0)) when P(X = 0) = 0; the formal condition ξ ≪ µ ensures ξ ⊗ν is well-defined.Source: §A.2 · pp. 28–29

Estimator conditions

• Estimator convergence requires regularity compatible with these operations, and practical transfer additionally depends on data support unless strong parametric assumptions are imposed.Smoothness assumptions can make computations involving Dirac sequences safe, but interventions cannot generally target values never observed without assumptions such as linearity.Source: §A.2 · pp. 29–31

Regular computations

• A standard-regular computation is a finite composition of identity or projection, ⊗-products, left-disintegrations, right-marginalizations, and transpositions.This notion reproduces standard IID results while applying to the paper’s more general measure-theoretic setting; compositions and anti-causal disintegrations are examples of regular functionals.Source: §A.2 · p. 30

Limits of computation

• The framework conjectures that generic kernels admit no further useful computations beyond regular ones, although infinite limits may be neither regular nor uniquely well-defined.This issue arises for limits of repeated kernel products, such as stationary distributions of time-series kernels.Source: §A.2 · p. 31

Structured kernels

• Regular kernel computations are defined, but their correspondence to operations on graphs of structured kernels remains unresolved in this section.The formalism therefore clarifies computation on general kernels without yet completing the graph-level interpretation for structured kernels.Source: §A.2 · p. 31

A.3 Graphical Operations

• Lemma A.17(a) establishes the central atom result, enabling c-component factorization, gluing, and extraction from backdoor-free families.Atoms decompose structured kernels into per-observable-node factors, and the kernel is a regular functional of its atoms.Source: §A.3 · pp. 31–33
• The framework distinguishes local subgraph computations, semi-local gluing from two compatible smaller graphs, and revealing hidden nodes using a graph with fewer latent variables.A more general revealing operation could make gluing on revealed parts genuinely non-local.Source: §A.3 · p. 31
• Simplification operations regularly compute kernels for suitable subgraphs by marginalizing hidden nodes and removing descendant-closed inner-node sets.A unique maximal removable set can be constructed by iteratively filtering nodes according to descendant chains.Source: §A.3 · pp. 31–32
• The full kernel is a regular functional of the jointly specified non-trivial c-components, extending the IID role of c-components in identifiability.C-components group observed nodes connected through hidden confounders, with associated latent ancestors included in the corresponding equivalence classes.Source: §A.3 · p. 34
• C-subgraph simplification yields regular functionals of the original kernel, while revealing reconstructs kernels from the full model and a c-component with fewer latent nodes.These operations support repeated decomposition and gluing, but the general computability of revealing remains subtle.Source: §A.3 · pp. 38–39

A.4 Algorithmic Construction

• SVSearch constructs target structural kernels by recursively gluing known kernels, remaining complete in the IID case although completeness is unknown generally.CGDecomp provides the complementary local decomposition into smaller c-subgraphs and regular functionals.Source: §A.4 · p. 39

A.5 Revealing Operations

• More general revealing operations appear substantially harder than gluing: missingness constraints and unavailable latent-variable information can prevent reconstructing desired kernels.The appendix presents an iterative c-subgraph revealing algorithm, but explicitly notes that failure to find a more general operation does not prove impossibility.Source: §A.5 · pp. 39, 41

B Details on Models and Observations

• The models-and-observations appendix supplies proofs and basic properties needed for identifying model properties from observations, including clarification of subspace intersections.In practice, shared value spaces make the required compositional domains clear.Source: B Details on Models and Observations · pp. 41–42

B.1 Results in the Main Text

• The shallow distribution exists uniquely, and an observable world realizes it through measurable parent functions and independent uniform noises while satisfying local Markov properties.These results rely on acyclicity and finite past; uniqueness is up to equality in distribution.Source: §B.1 · p. 42

B.2 Properties of Probabilistic Worlds

• Probabilistic-world properties establish single-level and multi-level transformations of structural graphs while preserving aligned mechanisms and enforcing latent-path graphical conditions.The multi-level construction can enlarge the graph and add copies of nodes aligned to singular kernels to satisfy the required graphical property.Source: §B.2 · pp. 43–45

C Details on the Extraction of Structured Kernels

• The structured-kernel extraction framework is extended to multi-level settings using finite local graphs with proper, pinned, and external nodes aligned to the underlying structural graph.In the single-level case, pinned nodes are absent and the local, observational, and underlying graphs agree.Source: C Details on the Extraction of… · p. 45, §C.1 · p. 46

C.1 Multi-Level Modifications

• Multi-level embedding families use injective, model-aligned mappings that include I-graph parents and proper edges, together with anchors, symmetries, applicability ranges, and singular-node assignments.Pinned nodes must be singularly embedded, while singular external nodes can be converted into pinned nodes, potentially making the family backdoor-free.Source: §C.1 · p. 46
• Multi-level modifications incorporate pinned, potentially unobserved but j-independent variables into datasets, direct identifiability, pinned kernels, associated structural graphs, and extracted knowledge sets.Backdoor-freeness and ancestral structures are adjusted for pinned and external nodes, while immediately observed variables and known interventions remain identifiable.Source: §C.1 · pp. 47–49

C.2 Embedding-Families and Attached Structure

• Larger graphs have larger latent sets and essentially larger ancestral structure, making attached structure monotonic and simplifying algorithmic search.The monotonicity results rely on sub-families, observed-node availability, and compatible ancestral paths.Source: §C.2 · pp. 50–51

C.3 Identifiability

• The framework deliberately separates causal identifiability—having suitably curated data for a query—from statistical estimation and estimator consistency.Directly identified kernels retain the information relevant to the model mechanisms and queries, so compatible estimators can be applied to the curated data.Source: §C.3 · p. 51
• Stationarity is unnecessary for systematic symmetry-based identifiability, although it can help as an a-priori assumption about a specific model realization.Using stationarity to structure the problem may instead shift and obscure the underlying difficulty.Source: §C.3 · p. 51

C.4 Identification from Families of Embeddings

• Valid datasets may use degenerate Xj, but the resulting limitation is observational support rather than a distinct multi-level statistical problem.If all embeddings yield the same value, observational support is restricted and Assumption 2.23 limits transfer conclusions; analogous issues can occur with non-X-degenerate datasets.Source: §C.4 · p. 52
• Backdoor-free families of embeddings make the associated kernel μ(Gobs(G, L)) δi-identifiable.The proof aligns each embedding with a structural graph, shows the observed graph forms compatible structural c-components, and establishes j-independent atoms for identification.Source: §C.4 · pp. 52, 54

C.5 Algorithmic Construction of Backdoor-Free Families

• Algorithm ExtractCS-R extends a decorated family to obtain non-unique c-connected backdoor-free families, from which basic queries are identifiable.By repeatedly decomposing structural c-components and applying Lemma 3.8, any basic query can be regularly computed from kernels associated with c-connected families.Source: §C.5 · p. 55
• The construction algorithms are intentionally non-unique and differ from the IID case by anchoring embeddings to one variable and lacking a general maximal-to-subgraph decomposition.Disconnected graph embeddings generally do not fit the formalism, and maximal c-components cannot generally be used as the starting point for extracting c-subgraphs.Source: §C.5 · p. 55
• ExtractCS constructs c-connected, backdoor-free embedding families by initializing direct embeddings, repeatedly absorbing relevant children, and stopping at convergence or k_max.ExtractCS-R attaches monotonic structure, absorbs hidden parent-chains and backdoor-path starts, while practical implementations can lazily enumerate query-matching families.Source: §C.5 · p. 56
• Smaller c-structures can yield better identification than the largest local c-structure when only those subgraphs are observed.The largest c-structure has the worst symmetry, while proper subgraphs may have smaller minimal latent sets and better identification results.Source: §C.5 · p. 57

D.1 Multi-Level Modifications

• In the multi-level case, structured queries include auxiliary indices, and their underlying basic query is identifiable when a regular functional has δ_i-identifiable inputs.The representation excludes queries whose targets are ancestors of observations, a weaker condition than disconnecting targets from observations in the I-graph.Source: §D.1 · pp. 57–58

D.2 Structured Representation of Basic Queries

• An abstract basic query has a corresponding structured query exactly when its targets do not cause observations and its external variables cannot be bypassed by directed paths.The proof constructs a finite embedded local graph from ancestors of the targets, aligns its mechanisms, pins observed context nodes, and verifies validity and model alignment.Source: §D.2 · p. 59

D.3 Identification of Queries

• The query formalism’s main technical step associates a structured graph with the realized-world distribution, yielding the kernel representation needed for identification.The resulting expression identifies µ with a conditional kernel involving the query outcome, external variables, pinned variables, and observed data.Source: §D.3 · pp. 60–61
• Theorem 2 establishes that a structured query and its underlying causal query are identifiable when its structured kernel is regularly computed from δi-identifiable components.The proof combines the realized-world connection with closure of δi-identifiability under regular functionals.Source: §D.3 · pp. 59, 61

D.4 Beyond Basic Queries

• Basic queries include conventional do- and soft-interventions while remaining more flexible and readily extensible.The framework can compile IID do-interventions, represent conditioned interventions, and stack additional reasoning through meta-queries.Source: §D.4 · p. 62
• Stationarity assumptions can support identification in time-series by estimating a finite stationary block and inserting it as a known kernel into query-identification formulas.This construction is described for stationary k-Markov time-series with suitably manageable missingness.Source: §D.4 · p. 62
• The formalism can represent intervention-induced stationary distributions as a fixed-point question whenever the resulting query has a Markov-kernel form.The framework supplies a kernel suitable for asking which stationary distributions exist after intervention.Source: §D.4 · p. 62
• Time-series queries can separate extracting a past kernel from specifying an intervention, enabling reasoning about soft or intervention-free effects even when stationarity or finite-past conditions fail.The example permits partial results and can estimate an invariant state from one query before using it as an intervention in a second query.Source: §D.4 · p. 62
• Nested queries extend structured-kernel reasoning by using graphical operations with disintegration nesting to align nodes with disintegrations of other queries.The paper proposes this for repeated ecosystem propagation in which one system selects, discards, or multiplies sub-systems generated by another.Source: §D.4 · p. 63

D.5 Examples for Multi-Level Queries

• The river-site example asks how water level Y changes when throughput X is enforced at an observed site, a nearby unobserved site sharing sediment, or an arbitrary unobserved site.These queries use the same mechanism Y_x,g but differ in how the query site connects to sediment and geometry contexts.Source: §D.5 · p. 64
• Identification requires increasingly broader data: Y1 needs many time-points at one site, Y2 needs observed geometry or many sites sharing sediment, and Y3 needs many sites.The freeness condition uses one Y sample per site, although unequal time-point counts require suitable reweighting.Source: §D.5 · p. 64
• Local queries are intrinsically more precise, but finite-sample estimator variance can favor broader queries because Y3, then Y2, can use more data than Y1.Y1 mainly reflects observation noise, whereas Y2 additionally includes geometry variation and Y3 also includes sediment variation.Source: §D.5 · p. 64
• Repeated outcomes are IID for Y1 but only exchangeable for Y2 and Y3 because the latter share a site-specific hidden geometry context.The formalism represents this through one shared Cgeom node with multiple Y children, distinguishing conditional IID from marginal IID.Source: §D.5 · p. 64
• The example exposes a limitation of gluing: learning a single-outcome structure with hidden Cgeom does not identify the joint multi-time-point Y2 distribution when the hidden context must be reused.The graphical query perspective makes this subtle incompatibility explicit.Source: §D.5 · p. 65
• Multi-level queries are formally distinguished by how their query components attach to the observational I-graph, a distinction unavailable in the disconnected single-level case.Figure 7 illustrates the three query structures for Y1, Y2, and Y3.Source: Figure 7 · p. 63, §D.5 · p. 65

E The IID Case

• The formalism is compared with established IID results to validate that its answers are sound.The paper reports that, although general completeness is not shown, the algorithms are complete in IID settings.Source: E The IID Case · p. 65

SCM formulation

• An SCM specifies IID endogenous variables, jointly independent exogenous noises, parent sets, measurable structural mechanisms, and uniquely solvable equations, with latents explicitly represented.The formulation assumes latent variables have no parents, at most two observed children, and parent mechanisms satisfy a minimality condition.Source: §E.1 · pp. 65–66

Interventions

• Interventions preserve exogenous noises and unaffected mechanisms, while hard interventions remove incoming edges and do-interventions replace mechanisms with constants.The intervened value must belong to the variable’s value space, and the corresponding graph is obtained by amputating incoming edges.Source: §E.1 · p. 66

Identifiability

• Causal effects are identifiable when P(Ỹ | do(Ẋ = x), Z̃) is uniquely computable from the observed joint distribution for every SCM inducing the graph.This generic definition does not rely on internal restrictions such as linearity and concerns all compatible SCMs producing the same observational distribution.Source: §E.1 · pp. 66–67

Hedge criterion

• An interventional distribution is identifiable if and only if the causal graph contains no hedge for the intervention and outcome sets.Hedges are defined using nested rooted c-forests satisfying the stated ancestry and sparse-subgraph conditions.Source: §E.1 · p. 67

ID-algorithm completeness

• The ID-algorithm is complete: whenever it fails to identify P(Ỹ | do(Ẋ = x)), it outputs a hedge, so failure exactly witnesses non-identifiability.This follows from the completeness theorem and the hedge criterion.Source: §E.1 · pp. 67–68

E.2 Translating the Standard IID-Setup

• IID models use dataset, variable, and infinitely many sample indices with permutation symmetries, making exchangeable samples conditionally IID given dataset identity.The viewport asymptotically observes variables in O across observed contexts, while variables in L remain unobserved.Source: §E.2 · p. 68
• SCM mechanisms become probability kernels, allowing an SCM and its intervention P(Ỹ | do(X̃ = x)) to be represented as an IID translated model.The translated model separates observational context α from intervention context β and aligns their local graphs with observational and interventional mechanisms.Source: §E.2 · pp. 68–69
• The translated query recovers the original intervention distribution: θ({V_i}_{i∈{β}×Ỹ×{*}}) = P(Ỹ | do(X̃ = x)).The intervention parameters x specify the singular intervention mechanisms in context β, while the query has an empty conditioning set.Source: §E.2 · p. 70
• In the IID case, direct identifiability reduces to knowledge of the observational distribution P and its conditional distributions, and regular-functional identification implies IID identifiability.Consequently, identifying the structured do-query identifies P(Ỹ | do(X̃ = x)) in the IID sense.Source: §E.2 · pp. 70–71

E.3 Single-Context with Empty Conditioning Set

• In the single-context setting, the ED-phase validates data-supported local graphs and enables assembly of a useful frontdoor-like strategy from partial results despite only one datapoint at the target site-time.It identifies separate X→Z_s and Z_s→Y_t components on asymptotically large datasets before the assembly phase glues them together.Source: §E.3 · p. 71
• The ED-phase outputs all c-fragments obtainable from μ(G_L, L) through repeated c-component decomposition and simplification, equivalently matching the decomposition result of the D-phase alone.The process terminates after finitely many steps because nontrivial decomposition and simplification reduce graph size.Source: §E.3 · p. 72
• The assembly phase uses c-fragment and c-forest relationships to construct rooted c-forests and extend them when a relevant c-fragment is absent.The supporting lemma states that c-fragments contain c-forests, c-forests in Gdo remain c-forests in G, and suitable rooted c-forests can be enlarged.Source: §E.3 · p. 72
• EDAIdentify is complete for single-context IID do-interventions with an empty conditioning set, returning identification exactly when no hedge exists.This is the stated IID completeness corollary, analogous to completeness of the id-algorithm.Source: §E.3 · p. 73
• For P(Y| do(X) = x), EDAIdentify returns a non-empty set if and only if no hedge exists for the queried intervention.The proof establishes the equivalence by showing that an empty algorithm output implies a hedge and that a hedge prevents identification by soundness and the hedge criterion.Source: §E.3 · pp. 73–74
• EDAIdentify and its gluing search terminate after finitely many steps because the query graph and available gluing elements are finite, while each gluing adds at least one node.The broader algorithmic construction also has finitely many forks because finite graphs contain finitely many c-components and subsets of L.Source: §E.3 · pp. 72, 74

E.4 Single-Context with Non-Empty Conditioning Set

• The apparent disagreement with [56] arises because this approach assumes no known or exploitable internal structure in mechanisms, whereas [56] implicitly does so through do-calculus Rule 2.Rule 2 allows replacing P(Y| do(X = x), M = m) with P(Y| do(X = x, M = m)).Source: §E.4 · p. 74
• Conditional effects are generally not identifiable, becoming identifiable when the intervention Xdo takes a single value.With non-singular interventions, conditioning on M can induce selection bias on Xdo, and confounding between X and M prevents estimating the required joint distribution.Source: §E.4 · p. 74
• Conditioning directly on intervened variables recovers do-calculus-style conditional queries without additional theory, because the corresponding Yx,m query is identifiable.This remains meaningful when Xdo is non-singular, although it does not reveal other internal structures affecting selection bias.Source: §E.4 · p. 75
• A second route extends regularity and computation rules so selected intervention or model kernels have trivial selection bias and admit regular functionals.For multidimensional or categorical X, independence of components relevant to M and Y can make the example query identifiable.Source: §E.4 · p. 75
• Thus, do-calculus combines causal rules with conclusions based on intervention structure, which this framework can represent by conditioning on interventions or constraining allowed computations.The latter option is more complex to implement but can provide insights for categorical or multidimensional variables.Source: §E.4 · p. 75

E.5 Standard Results for the Multi-Context, Experimental Case

• Selection diagrams represent multiple related SCM contexts by adding a selection variable S_v→v whenever the mechanism for v differs between paired systems.The framework extends from two contexts to source domains and a target domain, with observational and interventional distributions available across contexts.Source: §E.5 · pp. 75–76
• The paper revises the mz-shedge definition because its path requirement can contradict identifiability examples, while its inducedness clause is internally inconsistent when modifying only one context.It gives the example that P(Y | do(X = x)) is identifiable whereas P(Y, Z | do(X = x)) is not, and proposes an inductive mz-shedging formulation.Source: §E.5 · pp. 76–77
• The TRmz-algorithm is complete: when it fails, it produces graph-pair subgraphs containing a mz*-shedge, and the shedge criterion implies non-transportability.The paper notes a qualification concerning the cited completeness proof, but states completeness as the resulting corollary.Source: §E.5 · p. 78

E.6 Translating the Transportability Setup

• The mz-transportability setup has equivalent representations as different systems sharing properties or as copies of one system organized by selection variables in a multi-level structure.The shared-properties translation constructs an IID model with an infinite sample index, context/intervention-specific observations, and a single query dataset element.Source: §E.6 · p. 78
• Under the shared-properties translation, each observational context retains the shared causal graph or its intervention-modified graph, while the query context uses the graph intervened on X̃.The query graph, embedding classification, and identification definitions carry over from the single-context case.Source: §E.6 · p. 78
• The IID transportability model treats context as a hidden system property and encodes dataset-specific or shared context through auxiliary selection variables added to the I-graph.These variables are realized per dataset rather than per sample, and the construction extends the index set with a disjoint copy of the original variables.Source: §E.6 · p. 79

E.7 Multi-Context and Transportability

• EDAIdentify’s ED-phase finitely outputs all c-fragments in the relevant context graphs or do-intervention graphs, together with identification formulas.Different contexts correspond to disconnected components in the I-graph.Source: §E.7 · p. 80
• EDAIdentify returns a nonempty, sound identification strategy for P(Y | do(X = x)) if and only if no mz-shedge obstructs the corresponding transformed query.Extracted c-fragments can serve mz-transport queries when their nodes align with the same kernels as in the query-graph; experimental targets have additional structural restrictions that motivate conditions 2 and 3 of mz*-shedge.Source: §E.7 · p. 80
• The multi-level query-graph adds pinned auxiliary nodes for hidden system properties, requiring aligned kernels across target and source contexts to match during graph gluing.Experimental knowledge accounts for the additional mz*-shedge conditions governing pinned-node agreement and intervention structure.Source: §E.7 · p. 80

E.8 Mediation and Counterfactuals

• The query language identifies natural direct effects through the mediation formula without requiring counterfactual variables, including with additional observed confounders integrated out.The paper argues that some counterfactual difficulties arise from how practical questions are formulated, while questioning whether non-injective structural-mechanism counterexamples justify counterfactuals’ special status.Source: §E.8 · p. 81

E.9 Conclusion

• The formalism’s main purpose is to unify causal questions such as mediation and transportability while expanding the data structures and queries that can be formally represented.Soft-intervention generalizations are possible, but extending one setup is secondary to unifying apparently different causal questions in a shared language.Source: §E.9 · p. 81

F.1 Statistical Considerations

• Small, symmetric mechanism compounds may improve finite-data learning and robustness to structure misspecification, but complex estimators and large queries can harm statistical properties.Estimator choice must be propagated through compound computations in a statistically correct and efficient way.Source: §F.1 · p. 82

F.2 Selection Bias and Correlated Missingness

• Correlated missingness can be modeled with random observation indicators and contains selection bias as a special case, but its general treatment is beyond scope.Nested queries and enlarged backdoor-free graphs provide accessible special cases, although extraction can produce nested graphs that regular computation must handle.Source: §F.2 · p. 82

F.3 Structure Discovery

• Structure discovery is left for future work; conventional discovery applies under restricted IID or time-series priors, while invariant mechanisms may aid discovery in the general framework.Query estimates are identifiable when they do not depend on the representative chosen within a discovered Markov equivalence class.Source: §F.3 · pp. 82–83

F.4 Algorithmic Approach

• The algorithmic approach remains future work, with sketches based on embedding families, parent and child absorption, ancestral structures, and recursive sub-algorithms.Non-uniqueness and intermediate-result organization are the main practical challenges; simpler IID and multi-context transport cases appear more tractable.Source: §F.4 · pp. 83–85

G Mathematical Basics and Notation

• The paper develops customized measure-theoretic and probability-kernel results, culminating in novel graph-structured kernel results tailored to its formalism.The mathematical setup assumes standard Borel spaces and uses them to simplify the presentation of disintegrations and related results.Source: G Mathematical Basics and Notation · p. 85, §G.1 · p. 85

G.1 Measure-Theory

• The measure-theory review translates standard results into a uniform language over standard Borel spaces, including Borel, Polish, Radon, and probability-measure foundations.Finite Borel measures on Polish spaces are Radon, supporting the stated foundational framework.Source: §G.1 · pp. 85–86
• Under σ-finite Radon measures and countably generated measurable spaces containing singletons, disintegrations exist and are almost-everywhere unique; standard Borel joint measures therefore admit unique conditional probability kernels.For a probability measure on X × Y, the projection marginal determines a probability kernel representing the conditional law on Y.Source: §G.1 · pp. 86–87

G.2 Kernels and Basic Properties

• Probability kernels are measurable measure-valued mappings whose products and compositions encode joint distributions and causal mechanisms, while disintegration recovers factors and anti-causal conditioning, with composition generally not right-distributive.The anti-causal kernel depends on the joint distribution and both component mechanisms, whereas disintegration of confounded terms does not simply return the right-hand-side kernel.Source: §G.2 · pp. 88–91

G.3 Order Independent Operators

• Transposition provides a Borel-isomorphic reordering of kernel products, and generalized marginalization supports an order-independent treatment of multiple marginalizations and disintegrations.The section motivates this framework because the order of such operations is otherwise unclear.Source: §G.3 · p. 92
• G.3 defines unique disintegration operators from marginal and conditional kernels and establishes their compositional properties through universal properties and uniqueness.The construction proceeds inductively over subsets and supports combining and iterating disintegrations without dependence on the chosen decomposition order.Source: §G.3 · pp. 92–94

G.4 Sparsity and Contraction

• G.4 formalizes causal sparsity through contractions that wire kernels to parent arguments, while showing that hidden-confounding composition generally prevents recovering the original conditioned factors.The framework also derives transformations for commuting unconfounded products, trivial compositions, distributive composition, and argument behavior under anti-causal disintegration.Source: §G.4 · pp. 94–97

G.5 Symmetries

• The alternative formulation relies on orbit-sets rather than group-actions, removes rigidity, and remains strictly more general while Theorem 1 still applies and produces the same output.Group-actions remain useful as the standard approach to symmetries in physics and other natural sciences, offering intuitive descriptions and efficient encoding of models and prior assumptions.Source: §G.5 · pp. 97–98