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Symmetry-Informed Causal Partial Identification

Uzair Akbar, Zulfiqar Zaidi, Niki Kilbertus, Krikamol Muandet, Bo Dai

arXiv:2610.09230v1cs.LG

TL;DR

Partial-identification bounds can be near-vacuous, so the paper investigates known data symmetries as additional causal constraints. It operationalizes them as a shape constraint and through data augmentation as a change of measure, and reports sharper bounds under two canonical PI models. Its conclusions about misspecified symmetries are established for those two models, with broader applicability left for future investigation.

  • Problem

    Partial-identification bounds can be near-vacuous, and known data symmetries remain an underused source of domain knowledge for constraining causal effects.

    Source: §1 · p. 1

  • Method

    The paper uses known invariance as a causal-function shape constraint and as a change of measure implemented by symmetry-based data augmentation.

    Source: §1 · p. 1

  • Results

    The paper reports that both approaches sharpen bounds under two canonical PI models, with theoretical analysis and finite-sample experiments supporting the framework.

    Source: §1 · p. 1, §4.1 · pp. 7–8

  • Takeaways & Limitations

    Known data symmetries provide a natural, underused source of background knowledge for robust causal inference and partial identification.

    Source: §1 · p. 1

    The framework’s treatment of ε-misspecified symmetries is established for bounded-confounding and leaky-instrument models; additional models remain future work.

    Source: §6 · p. 10

Abstract

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Partial identification (PI) entails estimating bounds on causal effects by encoding different assumptions on data generation as a constrained optimization problem. Such bounds can suffice to inform policy decisions even if the causal effect itself is not identifiable. Often vacuous in practice, practitioners seek to exhaustively encode domain knowledge as additional constraints to make the PI bounds more informative. We introduce known data symmetries -- invariance of the causal effect under certain data transformations -- as a new source of constraints to inform PI. We operationalize this as a shape constraint on the causal function, and via a change of measure against which PI is posed using simple data pre-processing. Both approaches are shown to sharpen bounds under two canonical PI models. This is shown both theoretically for the population case, and via experiments in the finite-sample case. More broadly, our framework establishes data symmetries as a natural, underutilized source of background knowledge for robust causal inference.

1 INTRODUCTION

Partial identification bounds causal effects under assumptions, but can be near-vacuous. The paper proposes known data symmetries as an additional constraint and develops two ways to use them to sharpen bounds.

1 INTRODUCTION

  • PI can yield near-vacuous bounds, making credible domain knowledge important for restricting compatible data-generating processes.The bounds depend on the strength of the assumptions encoded in the optimization.Source: §1 · p. 1
  • Known data symmetries—properties unchanged by specified data transformations—are an underused source of constraints on candidate causal functions.For example, image content may remain unchanged under rotations, reflections, or translations.Source: §1 · p. 1
  • The paper operationalizes symmetries as a shape constraint for white-box solvers and as a change of measure using preprocessed data for off-the-shelf or black-box solvers.The change-of-measure approach is framed as a soft intervention that preserves the causal mechanism and graph up to relabeling while diluting confounding.Source: §1 · p. 1

2 PRELIMINARIES

The preliminaries distinguish prediction from causal estimation, introduce partial identification when point identification is unavailable, and define known symmetries as invariances of the causal function under data transformations.

2.1 STATISTICAL VS. CAUSAL INFERENCE

  • When treatment and unobserved noise are dependent, the conditional mean can be biased for the causal function, motivating causal inference and bounds when point identification is unavailable.Observed pretreatment covariates may provide auxiliary information to mitigate confounding bias.Source: §2.1 · p. 2

2.2 POINT IDENTIFICATION VIA INTERVENTION AND ADJUSTMENT

  • Interventions identify the causal target, while back-door adjustment or instrumental variables can recover it under their respective access and validity conditions.The IV route requires an unconfounded instrument with an exclusion restriction; its solution also requires completeness for uniqueness.Source: §2.2 · pp. 2–3, §2.3 · p. 3

2.3 PARTIAL IDENTIFICATION AND SENSITIVITY ANALYSIS

  • Partial identification bounds causal functions using all data-generating processes compatible with the observed data and assumptions, trading point identification for bounds that depend on those assumptions.The paper describes diverse constraints, including bounds on hidden confounding, robust-optimization neighborhoods, response-function models, and shape restrictions.Source: §2.3 · p. 3

2.4 KNOWN DATA SYMMETRIES AND INVARIANCE

  • A known symmetry is a transformation under which the causal function remains invariant; data augmentation samples such transformations and applies them to observed inputs.The paper frames this invariance as a shape constraint for sharpening PI and focuses its theoretical analysis primarily on the infinite-sample setting.Source: §2.4 · p. 4

3 A SYMMETRY SHAPE CONSTRAINT

The paper encodes known invariance as a feasibility constraint in a PI solver, while noting implementation barriers that motivate a data-only alternative.

3 A SYMMETRY SHAPE CONSTRAINT

  • A direct operationalization imposes an invariance-error constraint, with ε serving as a sensitivity parameter.Other invariance functionals or an invariant hypothesis class are also possible, though the latter may not support sensitivity parameters.Source: §3 · p. 4
  • The constraint can be incorporated into deep-learning PI solvers, but requires white-box access and may be expensive and unstable for non-convex, high-dimensional problems.Differentiability and Monte Carlo evaluation permit regularization in augmented-Lagrangian or gradient-based solvers.Source: §3 · p. 5
  • These operational limitations motivate the paper’s second approach, which acts on the data rather than the solver.Source: §3 · p. 5

4 A CHANGE OF MEASURE UNDER SYMMETRY TRANSFORMATIONS

The paper uses symmetry transformations to change the measure under which partial identification is posed, while preserving validity across broad families of standard solvers. It shows that this can sharpen bounds under observable conditions and develops a robust pad-and-intersect procedure, then uses the transformation itself as an instrument to impose symmetry within IV solvers.

4 A CHANGE OF MEASURE UNDER SYMMETRY TRANSFORMATIONS

  • Data augmentation acts as a soft intervention that preserves the causal mechanism and graph up to relabeling while diluting confounding, allowing broad families of valid PI constraints to remain valid.The change of measure reweights candidate evaluations and can prune previously feasible functions.Source: §4 · pp. 5–6

4.1 PARTIAL IDENTIFICATION WITH DATA AUGMENTATION

  • With T-invariant causal effects, post-DA bounds are exactly valid when the baseline set is sufficiently approximately valid; dilution lets the inherited confounding budget accommodate the true bias.The validity result relies on the simulated intervention, not on conditions over the unobserved bias direction.Source: §4.1 · p. 7
  • Bounds tighten when variance inflation is sufficient to offset noise inflation and mean-shift penalties, captured by the observable condition Ω≤1.Noise inflation and mean-shift terms expand the set; the soft intervention keeps the true bias within budget as other candidates are pruned.Source: §4.1 · pp. 7–8
  • For ε-misspecified symmetry, ε-padding the post-DA intervals and intersecting them with the baseline set preserves validity and dominates the baseline in sharpness and worst-case error.The intersection guarantee holds even when post-DA bounds change arbitrarily.Source: §4.1 · p. 8

4.2 SYMMETRY CONSTRAINT WITH T -AS-IV

  • T-as-IV imposes symmetry inside an off-the-shelf IV solver by adding the random transformation T to the instrument, although T-invariance makes this instrument incomplete.The incomplete instrument still constrains partial identification.Source: §4.2 · p. 8
  • On the optical-device benchmark, symmetry-informed bounds reduce interval width and worst-case error over baseline while maintaining validity across principal-component queries and a radial sweep.The benchmark augments images with flips, 90° rotations, and sum-zero translations.Source: Figure 5 · p. 9

5 EXPERIMENTS

Across the experiments, symmetry-informed PI sharpens bounds while retaining validity, with data augmentation useful where the invariance shape constraint cannot be implemented. The examples range from optical-device and cigarette-demand data to do-MNIST, with ablations testing robustness.

Optical device

  • For optical-device data, symmetry-informed bounds remain valid while reducing width and worst-case error, sometimes by an order of magnitude.The transformations are flips, 90° rotations, and sum-zero translations, which should not affect irradiance.Source: §5 · p. 9

US cigarette demand

  • For US cigarette demand, symmetry makes the neighbor-price effect clearly positive unless the IV is heavily leaky, suggesting cross-border shopping.The assumed symmetry scales all prices and income together; Figure 6 varies confounding and IV violation budgets for state and neighbor price elasticities.Source: §5 · p. 9, Figure 6 · p. 9

do-MNIST

  • On do-MNIST, DA reduces average width by 19.08% and worst-case error by 34.81%, while T-as-IV reduces them by 29.05% and 47.10%.The shape constraint is infeasible with CopSens's frozen, non-invariant feature extractor; coverage dips by 0.40% for DA and 1.31% for T-as-IV.Source: §5 · pp. 9–10

Ablations

  • Under deliberately misspecified symmetry, ε-padding restores coverage once ε ≥ ε⋆, while intersecting with the baseline set prevents regression.Improvements persist with symmetry violations as large as confounding itself, up to 16× over-padding, without budget recalibration, and across sample sizes.Source: §5 · p. 10

6 DISCUSSION

The discussion presents data augmentation as a practical alternative when an invariance shape constraint is difficult to implement, while identifying unresolved scope under misspecified symmetry. It also outlines broader potential applications and future extensions.

6 DISCUSSION

  • The main limitation is that DA's composability with disparate PI solvers under ε-misspecified symmetry remains unclear; analysis covers bounded confounding and leaky IVs.For leaky IVs, the appendix states that tolerance to misspecified symmetry is possible when the IV violation budget is positive.Source: §6 · p. 10, B MISSPECIFIED DA UNDER IV CONSTRAINTS · p. 18
  • When the invariance shape constraint is difficult to implement, practitioners can use DA; ε-padding is proposed when symmetry may be misspecified.The conclusion recommends established domain symmetries as part of the PI toolkit and describes DA pre-processing as effortless to integrate.Source: §6 · p. 10
  • DA may extend beyond the studied models to other bounded-confounding models and, under well-specified symmetry, to graphical constraints and other causal settings.The discussion names MSM, Wasserstein, total-variation, and f-divergence models, and describes applications involving latent IVs and proxy variables.Source: §6 · p. 10
  • Unlike prior symmetry work on identifiable targets, this paper studies PI when point identification itself fails.The discussion also distinguishes prior additive-Gaussian DA work on non-identifiable point estimation as fragile in general.Source: §6 · p. 10
  • Future work targets general causal queries, response-function equi-variance, latent confounding under Wasserstein misspecification, and applications in genomics and medicine.Source: §6 · p. 10

B MISSPECIFIED DA UNDER IV CONSTRAINTS

  • With positive IV violation budget γz, the appendix permits ε-misspecified symmetry through adjusted bounds or a recalibrated budget.The recalibrated budget must accommodate both IV leakage and symmetry misspecification; an orthogonality assumption can tighten it.Source: B MISSPECIFIED DA UNDER IV CONSTRAINTS · p. 18

C SHAPE CONSTRAINT VS. CHANGE OF MEASURE

The section compares invariance shape constraints with data augmentation under compact-group and relevance/completeness assumptions. In that setting, their identified sets share a center and invariant slice, with a dominance hierarchy for sharpness and worst-case error.

C SHAPE CONSTRAINT VS. CHANGE OF MEASURE

  • The comparison assumes compact-group DA, which makes the post-DA distribution invariant and the conditional-expectation operator an orthogonal projector onto the invariant subspace.Relevant or complete DA requires that only invariant directions have zero data-averaged response across augmentations.Source: C SHAPE CONSTRAINT VS. CHANGE OF MEASURE · p. 19
  • Under the stated assumptions, both routes yield identified sets centered at the post-DA ERM and restricted to the invariant function slice.The invariance-constrained set is the baseline PI set intersected with the invariant class; DA uses the post-DA set, optionally with T-as-IV.Source: C SHAPE CONSTRAINT VS. CHANGE OF MEASURE · p. 19
  • The shape-constrained set has a smaller radius than the DA-as-IV set, with equality when ρ = 1; the same dominance hierarchy holds for sharpness and worst-case errors.This comparison assumes an invariant target, no observed covariates, zero ε and γz, and a valid baseline PI set.Source: C SHAPE CONSTRAINT VS. CHANGE OF MEASURE · p. 19

D IMPLEMENTATION DETAILS

The implementation uses a common finite-sample PI program, with budgets relaxed to account for sampling variability. It applies different solvers and preprocessing choices across experiments, and reports narrower intervals and lower worst-case error than baseline in the simulation figure while maintaining validity.

D IMPLEMENTATION DETAILS

  • The empirical PI program imposes bounded-confounding, leaky-IV, and invariance constraints, with tolerances adjusted for sampling error.The hypothesis class uses a parameterized link and basis features; the implementation uses QCQP for identity-link experiments and SLSQP with five multistarts for do-MNIST.Source: D IMPLEMENTATION DETAILS · pp. 20–21
  • Finite-sample budgets are relaxed so the target satisfies each empirical constraint with asymptotic probability at least 1 − α/3; a union bound gives overall validity of at least 1 − α.The experiments use α = 0.05 and retain n as the original dataset size because augmented copies are dependent.Source: D IMPLEMENTATION DETAILS · p. 21
  • The finite-sample tolerances use asymptotic normality of empirical constraint statistics and population budgets to choose relaxed budgets.The passage specifies a noncentral chi-square distribution for the calculation; the invariance budget requires no finite-sample adjustment because it is assumed to hold almost surely.Source: D IMPLEMENTATION DETAILS · p. 21
  • DA constructs an augmented dataset by transforming each treatment and pairing each transformed observation with its original outcome and the instrument augmented by the transformation.Post-DA budgets are recalibrated for attenuated confounding and symmetry misspecification, then swept in the ablations.Source: D IMPLEMENTATION DETAILS · p. 21
  • Across simulation queries along the first two principal components of Var(X), interval width and worst-case error decrease over baseline while validity is maintained.Figure 8 displays these metrics across the two query directions.Source: Figure 8 · p. 22

E.1 EXPERIMENT SETUP

The experiments evaluate symmetry-informed partial identification across simulation, optical-device, cigarette-demand, and do-MNIST settings. The setups encode different symmetries and confounding structures, and do-MNIST also exposes a feasibility limitation of the shape constraint.

Simulation

  • In simulation, augmentation varies X only along directions in null(θ⋆), making the target exactly invariant without identifying θ⋆ absent observations of U.The construction uses A as a stand-in for domain knowledge of plausible symmetries.Source: §E.1 · p. 22

Optical device

  • The optical-device benchmark contains 12 datasets of 3 × 3 pixel images and photodiode voltages, with a withheld latent U used only to estimate the target.Augmentations combine flips, 90° rotations, and zero-sum translations that preserve the pixel sum; datasets #6 and #11 are excluded from the aggregate ablations.Source: §E.1 · p. 22

US cigarette demand

  • For US cigarette demand, symmetry assumes demand is unchanged when prices and income scale together, while state and neighboring-state prices are confounded.The data cover 48 contiguous states and DC from 1970–2019; income and CPI are assumed exogenous and, with cigarette tax, form the instrument set.Source: §E.1 · p. 23

do-MNIST

  • The do-MNIST setup creates color-confounded labels and applies image transformations, with the target ATE distinguished from the color-tracking ERM.The symmetry is exact, while ε = 0.0475 is used as finite-sample slack; 5% of augmented rows remain untransformed.Source: §E.1 · p. 23
  • On do-MNIST, the invariance shape constraint is infeasible for reasonably small ε because the frozen ERM features are not invariant.Unfreezing earlier layers would restore feasibility, but at the cost of vacuity and compute; data augmentation does not require solver access.Source: §E.1 · p. 23

E.2 ABLATIONS

The ablations report sharper symmetry-informed bounds across sample sizes and examine validity, robustness, recalibration, and augmentation size. The accompanying technical results describe the change in estimation geometry and the validity and informativeness arguments for data augmentation.

E.2 ABLATIONS

  • Symmetry-informed PI gives sharper bounds across sample sizes; post-DA sets attain coverage at smaller confounding budgets than baseline.With symmetry misspecification, bounds improve on simulation and optical device up to ε = 16ε⋆; cigarette-demand improvements are reported at ε = ε⋆ for the intersected bound. Gains persist without recalibration, while increased augmentation size has dataset-dependent effects.Source: Figure 10 · p. 24, Figure 12 · p. 25, Figure 13 · p. 25

TECHNICAL PRELIMINARIES

  • The change in estimation geometry decomposes into a positive variance-shift operator and a rank-at-most-one mean-shift term.The mean-shift term has trace D²; the variance-shift operator captures the geometry change between the original and augmented distributions.Source: TECHNICAL PRELIMINARIES · p. 27

F.1 PROOF OF THM. 1—VALID BOUNDS WITH DA

  • Theorem 1 states that post-DA bounds remain valid under a condition on baseline validity and the data-augmentation setting.Equality in the proof’s bound comparison holds precisely when the target belongs to the baseline PI set.Source: §F.1 · p. 28

F.2 PROOF OF THM. 2—INFORMATIVE BOUNDS WITH DA

  • Theorem 2’s proof decomposes the difference in average error into centroid drift and a set-size term.The proof also treats the interval case separately using a uniform distribution on a centered interval.Source: §F.2 · p. 29
  • For a uniform law over a centered ball, the proof reduces average errors to second moments and connects pre- and post-DA sets through the shift operator.The interval argument uses representers and the corresponding uniform-interval case.Source: §F.2 · p. 29, §3 · p. 29

F.3 PROOF OF THMS. 3.A AND 3.B—ROBUST BOUNDS WITH DA

The robustness proof establishes validity for ε-padded post-DA intervals and gives a condition for padded T-as-IV intervals. The accompanying propositions characterize estimation with augmentation and the geometry underlying sharper bounds.

F.3 PROOF OF THMS. 3.A AND 3.B—ROBUST BOUNDS WITH DA

  • ε-padded post-DA intervals remain valid for every ε ≥ 0, while padded T-as-IV intervals are valid under a sufficient condition.The T-as-IV argument uses a surrogate target within ε of the true target; a conservative alignment assumption yields the stated sufficient condition, while an uncorrelated-misspecification case permits a larger budget.Source: §F.3 · p. 30

F.4 PROOF OF PROP. 1—ESTIMATION WITH DA

  • The estimation proposition expresses the excess-risk difference using the bias direction, a variance-shift quantity, and pre-DA excess risk.The proof identifies the pre- and post-DA biases with conditional expectations of the confounder under the respective distributions.Source: §F.4 · p. 31

F.5 PROOF OF PROP. 2—SHARPER BOUNDS WITH DA

  • The sharp-bounds proof represents the post-DA set as an ellipsoid whose semi-axes scale with the square roots of the shift-operator eigenvalues.The volume comparison reaches equality when those eigenvalues coincide, equivalently when the shift operator is a positive scalar multiple of the identity.Source: §F.5 · p. 32
  • The proof also compares average interval widths by expressing them through representer norms and averaging over X.It relates the width comparison to the decomposition of the shift operator into variance- and mean-shift terms.Source: §F.5 · p. 32

F.6 PROOF OF PROP. 3—SHAPE CONSTRAINT VS. CHANGE-OF-MEASURE

Under the stated assumptions, the shape-constrained and change-of-measure sets share a center and invariant slice, allowing their radii to establish a dominance hierarchy. The finite-sample proof separately derives a coverage claim for enlarged budgets.

F.6 PROOF OF PROP. 3—SHAPE CONSTRAINT VS. CHANGE-OF-MEASURE

  • The shape-constrained set is contained in the T-as-IV set, which is contained in the post-DA set, yielding the same hierarchy for sharpness and worst-case error.All sets share a center and the invariant slice; comparing their radii establishes the inclusions.Source: §F.6 · p. 33

F.7 PROOF OF PROP. 4—FINITE-SAMPLE BUDGETS

  • The finite-sample proof establishes a coverage claim by rewriting the empirical constraint as a Gaussian-ball probability and bounding a noncentral χ² quantile.The population budget is enlarged by a positive term under the stated setup; the figure also depicts filtering baseline models before or after mapping to the hypothesis set as equivalent.Source: §F.7 · p. 34, Figure 16 · p. 35

F.8 PROOF OF PROP. 5—SHARP BOUNDS WITH STRUCTURAL-FUNCTION CONSTRAINTS

The proof establishes that function-level predicates yield sharp constrained identified sets, then shows that data augmentation (DA) acts as a soft intervention, preserving an invariant structural equation while attenuating confounding.

F.8 PROOF OF PROP. 5—SHARP BOUNDS WITH STRUCTURAL-FUNCTION CONSTRAINTS

  • For predicates on structural functions, the constrained identified set is sharp and equals the baseline candidate set intersected with the predicate-constrained set.The same identity holds for any finite collection of such predicates.Source: §F.8 · p. 35
  • The invariance-shape and leaky-IV models are sharp: each constrained set equals the intersection of its baseline set with the corresponding constraint set.These are the invariance constraint in §3 and the IV constraint in Asm. 3.Source: §F.8 · p. 35

F.9 PROOF OF LEM. 1—SOFT INTERVENTION WITH DA

  • Under an invariant target function, DA yields the post-augmentation structural equation Y = h⋆(eX) + U + ξ, with ξ independent of (U, eX, eZ).The lemma characterizes DA as a soft intervention.Source: §F.9 · p. 36
  • DA transfers Markov structure to the relabeled graph eG when X is replaced by eX and X has no descendants other than Y.The stated condition is NonDesc_G(X) = V \ {X, Y}.Source: §F.9 · p. 36
  • Under Asm. 1a, DA attenuates confounding, with the post-augmentation bias determined by the conditional mean of U given eX.The proof gives ||b̃||² = Var(E[U | eX]) ≤ Var(E[U | X]) = ||b||².Source: §F.9 · p. 36

F.10 PROOF OF LEM. 2—CHARACTERIZING THE IDENTIFIED SET4

These proofs characterize the baseline identified set and querywise intervals, then establish results on shift decomposition, error comparisons, DA projection, and average representer norms.

F.10 PROOF OF LEM. 2—CHARACTERIZING THE IDENTIFIED SET4

  • Under Asm. 2 and 1a or 1c, the PI set is a covariance ball centered at h_erm with radius σ√γ, where γ = γxγy; its querywise image is an attained interval.Every point in the ball has a Gaussian certificate, although the observed distribution need not be Gaussian.Source: §F.10 · pp. 37, 39

F.11 PROOF OF LEM. 3—SHIFT DECOMPOSITION

  • Under Asms. 1b–1c, S decomposes into positive-definite K and positive-semidefinite δ*δ, whose rank is at most one and trace is D².Source: §F.11 · p. 40

F.12 PROOF OF LEM. 4—AVERAGE ERROR AS A WORST-CASE SURROGATE

  • For squared loss on a bounded interval, worst-case error is at most 4 times average error, with equality at offset d = w/3.Source: §F.12 · p. 41

F.13 PROOF OF LEM. 5—REYNOLDS PROJECTION WITH DA

  • DA's conditional expectation is the L2(X)-orthogonal projection onto invariant functions; for an invariant target, it projects baseline ERM and bias to their post-DA counterparts.Source: §F.13 · p. 42

F.14 PROOF OF LEM. 6—AVERAGE REPRESENTER NORMS

  • The average-representer calculation expands query representers in an orthonormal basis and converts the resulting sum to tr(S).Source: §F.14 · p. 43

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