ProximalFM: Amortized Proximal Causal Inference under Hidden Confounding

1 Introduction

• Proximal learning uses proxy variables associated with hidden confounders, but nonparametric estimation requires solving ill-posed integral equations and can demand intensive, dataset-specific optimization.The introduction presents this as the practical estimation challenge motivating an amortized alternative.Source: §1 · p. 2
• PROXIMALFM is introduced as a proximal causal foundation model that uses proxies without dataset-specific training at deployment and estimates effects in one forward pass.The authors describe it as a Bayesian solution to an ill-posed inverse problem without hyperparameter tuning.Source: §1 · p. 2
• Its structural prior enforces proximal assumptions, and Monte Carlo integration supplies CATE targets that regularize the amortized estimation problem.The prior determines the inductive bias used by PROXIMALFM.Source: §1 · p. 2
• The evaluation spans semi-synthetic proximal benchmarks with real-world covariates and controlled causal mechanisms, plus an initial evaluation on the Causal Chambers physical testbed.The physical testbed is described as controlled.Source: §1 · p. 2

1.1 Related work

• Reviewed causal foundation models target backdoor or unconfoundedness, IV, and frontdoor settings, while a foundation model designed for proximal inference remains absent.The cited IV partial-identification methods typically do not exploit multiple proxies or completeness assumptions.Source: §1.1 · pp. 2–3, §1 · p. 2

2 Proximal causal inference

• The observed covariates are partitioned into baseline X, treatment-confounding proxy Z, and outcome-confounding proxy W, with both proxies informative about latent U.Z may relate directly to treatment, while W may directly shape the outcome.Source: §2 · p. 3

2.1 Identifying assumptions

• Identification assumes latent exchangeability, proxy exclusion restrictions, latent positivity, and completeness of the proxies for variation in U.The exclusion restrictions separate Z's relationship to treatment from W's relationship to outcome.Source: §2.1 · p. 3
• Completeness requires conditional expectations of functions of U given Z, and of functions of Z given W, to vanish only for zero functions.These two conditions are stated separately in the assumptions.Source: §2.1 · p. 3

2.2 Classical and modern estimation via bridge functions

• An outcome bridge h0 satisfies a conditional moment equation, and the proximal g-formula identifies CATE by averaging its treatment-level difference over W given X.The effect is conditional on X alone, rather than on the proxies.Source: §2.2 · p. 4
• P2SLS is efficient under strict linear specifications, while flexible kernel, neural, and doubly robust methods address more complex mechanisms but bring tuning, retraining, or kernel-computation burdens.The cited hurdles motivate an amortized alternative that estimates Q(x) directly from observational context.Source: §2.2 · p. 4

3 ProximalFM

• PROXIMALFM predicts a posterior distribution over CATE from observational context and query covariates in a single forward pass, without explicitly solving an ill-posed inverse problem.Pretraining on datasets from the proximal prior amortizes dataset-specific Bayesian inference.Source: §3 · p. 4, §3.2 · p. 6

3.1 A synthetic prior for proximal identification

• The synthetic prior varies dimensions, nonlinear mechanisms, feature types, noise, confounding strength, overlap, and treatment-effect heterogeneity across SCMs.Every sampled SCM is designed to satisfy proximal identification conditions.Source: §3.1 · p. 5
• The prior cannot guarantee completeness for arbitrary nonlinear mechanisms; instead, it heuristically requires W and Z dimensions and cardinalities to exceed U's.Completeness is a distribution-level condition that is untestable in observational data.Source: §3.1 · p. 5
• Rather than train on ITEs, whose distribution depends on an unobserved joint potential-outcome law, the method estimates each query's CATE by repeated downstream SCM draws at fixed x_i.The Monte Carlo mean is unbiased for Q(x_i), and its estimated sampling variance decreases as K grows.Source: §3.1 · p. 5

3.2 Model & objective

• The model encodes structural roles for X, Z, and W, embeds factual outcomes according to treatment, and uses only query x because proxy conditioning would change the CATE target.The query prediction targets the posterior over CATE rather than an individual-level ITE distribution.Source: §3.2 · p. 6
• A noise-aware distributional loss accounts for Monte Carlo target error, and inference returns the full CATE posterior plus its posterior-mean point estimate.The model blurs the predictive distribution to correct for estimated Monte Carlo variance.Source: §3.2 · pp. 6–7

3.3 From proximal identification to Bayesian consistency

• Under proximal identifying conditions, the oracle Bayesian posterior concentrates on the true CATE as context grows, and its mean converges to that value.The trained model's posterior-mean predictions inherit consistency if their approximation error relative to the oracle also vanishes.Source: §3.3 · p. 7

4.1 Evaluation setting

• The comparison includes proximal estimators, causal foundation models assuming no unmeasured confounding, and conventional S-, T-, and X-meta-learners.Backdoor methods are evaluated with X alone or with (X, W, Z) followed by proxy marginalization; neither is generally identified as Q(x) under proximal assumptions.Source: §4.1 · p. 7
• The 288 evaluation instances cross linear and nonlinear mechanisms, confounding strength, and proxy informativeness, using real-world features for X and U.The semi-synthetic design evaluates 12 datasets with three replicates in each of eight settings.Source: §4.1 · p. 7

4.2 Empirical results

• Across eight settings, PROXIMALFM is competitive, with its clearest advantage at larger contexts under high confounding and low proxy reliability.Figure 4 summarizes median CATE nPEHE across benchmark episodes; lower is better, with the dotted line marking the constant-ATE baseline.Source: §4.2 · p. 8, Figure 4 · p. 8
• Beyond nctx = 2 048 in the nonlinear, high-confounding, low-reliability setting, PROXIMALFM improves little and remains near the constant-ATE baseline, while P-learner continues improving.Limited additional training on contexts up to 5,000 observations yields small improvements at large context sizes, consistent with limited large-context training support contributing to the plateau.Source: §4.2 · p. 8
• At n = 2 048, PROXIMALFM lies on a favorable accuracy-runtime frontier, achieves among the strongest CATE ranks, and requires no hyperparameter selection.Moving inference from CPU to MPS further reduces runtime without changing predictions; Figure 5 plots median runtime against median paired CATE nPEHE rank.Source: §4.2 · p. 8, Figure 5 · p. 9
• On semi-synthetic data, the posterior becomes increasingly overconfident as context grows, and its standard deviation continues to decrease beyond the maximum pre-training context size.This undercoverage under distribution shift contrasts with coverage close to nominal on datasets sampled from the pre-training prior.Source: §4.2 · pp. 8–9
• At nctx = 2 048, PROXIMALFM outperforms proxy-marginalized CausalPFN throughout the proxy-information sweep and proxy-adjusted OLS in most settings; OLS catches up when proxies nearly reveal U.Well-specified proximal 2SLS provides a reference for what matching the estimator to this mechanism can achieve.Source: §4.2 · p. 9
• On Causal Chambers, backdoor meta-learners outperform proximal learners, including PROXIMALFM, in ATE estimation.The authors interpret this cautiously because PROXIMALFM targets CATE, while the benchmark covers twelve configurations of one one-dimensional system at 1,000 observations each.Source: §4.2 · p. 9

5 Conclusion

• PROXIMALFM amortizes estimation of CATE under proximal assumptions from synthetic episodes, avoiding a separate ill-posed bridge-equation solve for each dataset.Its prior supplies regularization, and the posterior also provides uncertainty estimates, though credible intervals can be overconfident beyond the training prior.Source: §5 · p. 9

A.1 Setup and identification

• CATE-identifiability means the CATE is a measurable function of the observational law, while CEPO-identifiability separately requires this for each treatment-specific conditional mean.CEPO-identifiability implies CATE-identifiability, but the converse need not hold because unidentified components can cancel in the difference.Source: §A.1 · p. 15

A.2 Consistency of the oracle CATE posterior

• Under Assumptions 5–6, CATE-identifiability at a fixed query is equivalent to oracle posterior consistency, and the exact CATE posterior concentrates under these conditions.The oracle posterior mean converges to the true CATE for prior-almost every SCM.Source: §A.2 · p. 16

A.3 Proximal identification

• When proximal assumptions, including completeness and bridge existence, hold, the conditional proximal g-formula identifies the CATE from the observational law.The formula represents the CATE as the conditional expectation of the outcome-bridge contrast given X = x; the trained model need not estimate the CEPOs.Source: §A.3 · p. 16

A.4 Application to PROXIMALFM

• Monte Carlo potential-outcome labels have the oracle posterior mean as their regression target at finite K, but exact full-posterior recovery needs an identifiable measurement-error model or vanishing Monte Carlo error.The stated theory does not establish exact recovery by the finite-K noise-aware objective or its combination with a mean penalty.Source: §A.4 · pp. 16–17

B.1 Step-by-step procedure

• Synthetic datasets share one sampled SCM across units and replicates; each unit’s covariates are replicated K times with fresh noise to estimate its conditional potential-outcome means and CATE.One replicate per unit supplies the observed tuple, while the CEPO targets are used only for training and are not observed by the transformer.Source: §B.1 · pp. 18–19
• Completeness cannot be guaranteed for arbitrary nonlinear mechanisms, so the prior uses numeric and categorical proxy-profile heuristics instead.The construction enforces other proximal conditions through mechanism restrictions, independent noises, and clipped treatment propensity.Source: §B.1 · p. 19

B.2.1 Parameters of the prior

• The prior varies SCM dimensions, nonlinear mechanisms, feature types, noise, confounding, treatment overlap, and effect heterogeneity across datasets.Structural parameters fix dataset shape, fixed hyperparameters are shared across the prior, and sampled hyperparameters vary per dataset.Source: §B.2.1 · p. 21

B.2.2 Dimensions & dimension heuristic

• The triangular confounder-dimension law and conditional-uniform proxy dimensions yield a linear-ramp proxy marginal, unlike the ceiling-heavy marginal induced by uniform confounder dimensions.For m = 10, the probability of proxy dimension 10 is 0.18 under the triangular law versus 0.29 under the uniform law; the triangular law also lowers mean confounder and proxy dimensions.Source: §B.2.2 · p. 24, Figure B.2 · p. 24

B.2.3 MLP generation, noise & edge sampling

• Proximal mechanisms compose random MLPs, with a sampled U-derived input share controlling confounding independently of U’s dimension.Fresh per-node noise blocks keep noise proportional to signal, and sampled column masks remove potential X and proxy-parent inputs without adding dependencies. Output diagnostics show channel contributions rise with their knobs; their dependence estimates are lower bounds because inputs are supersets, and unique-signal attribution understates U when its proxy shares information.Source: §B.2.3 · pp. 24–26, Figure B.4 · p. 26

B.2.4 Post-process: categorisation, outlier and scaling

• ≈0.70 vs. ≈0.60: categorical-completeness enforcement yields a higher proxy numerical-column fraction, while proxy category counts deepen with U’s category demand.The mean proxy category count rises from ≈12 at 2 categories to ≈19 in the 30+ bin under enforcement, versus ≈10 without it; enforcement deepens categories rather than increasing their share.Source: §B.2.4 · p. 27, Figure B.7 · p. 29

B.2.5 Treatment assignment

• Treatment assignment excludes W by construction, standardises mechanism logits, then uses per-dataset shift and scale knobs before a clamped Bernoulli draw.The shift changes marginal treatment prevalence, the scale controls assignment sharpness, and clamping propensities to [10^-3, 1 − 10^-3] ensures nonzero probability for both arms.Source: §B.2.5 · p. 30

B.2.6 Potential outcomes & heterogeneity remix

• The heterogeneity dial γ linearly interpolates unit effects between their raw values and the dataset ATE while preserving the ATE.Delivered effect dispersion tracks γ, and corr(Y0, Y1) approaches 1 as γ approaches 0.Source: §B.2.6 · p. 31, Figure B.9 · p. 32

B.2.7 Monte Carlo CATE supervision

• Monte Carlo CATE supervision averages paired potential-outcome effects across K downstream draws, with standard error decreasing as K^-1/2.The covariate row stays fixed while latent confounders, proxies, and outcome noise are redrawn; K changes target precision and generation cost, not the estimand.Source: §B.2.7 · p. 32

B.2.8 Visualize the prior

• The prior galleries visualize feature marginals, within-block dependence, and selected cross-block relationships, including proxy association through shared U.The cross-block examples include U–W, U–Z, W–Z, and U–propensity relationships.Source: §B.2.8 · p. 32, Figure B.10 · p. 33, Figure B.11 · p. 34

B.3.1 Overlap

• 84.0% of rows fall within [0.05, 0.95], while the median dataset still has 13.6% outside that practical-overlap band.Only 42.6% of datasets contain any row at the propensity clamp, and the median dataset contains none.Source: §B.3.1 · p. 36, Figure B.13 · p. 36

B.3.2 Exclusion restrictions

• The proxy exclusion restrictions hold by construction because the DAG omits the prohibited proxy-to-outcome and proxy-to-treatment links.The audit uses cross-fitted incremental R2 checks and positive controls that drop U from the conditioning set to test whether the regressions can detect predictive leakage.Source: §B.3.2 · p. 36
• Exclusion checks remain near zero, while positive-control dependence correlates with U-dependence at 0.375 on the Y side and 0.415 on the A side.The checks stay near zero regardless of confounding strength, whereas controls rise for datasets with greater U-dependence.Source: Figure B.14 · p. 37, §B.3.2 · p. 37

B.3.3 Latent exchangeability and confounding strength

• Errors above 0.1 sd(Yobs) occur in 24.1% of unadjusted datasets versus 10.7–10.8% with either oracle adjustment.Median errors are 0.041 unadjusted and 0.025 for oracle-(X, U, W); backdoor-X error is elevated when the delivered U share of fA exceeds 0.495, with median error 0.045–0.046 versus 0.028–0.041 in lower-share rows, while oracle error shows no comparable pattern. Latent exchangeability holds by construction, but these checks do not test conditional independence or establish finite-sample unbiasedness.Source: §B.3.3 · p. 38, Figure B.15 · p. 39, Figure B.16 · p. 39

B.3.4 Proxy relevance

• Weakest-column relevance rises with slack (ρ ≈0.28) and falls with latent dimension (ρ ≈−0.26), but is nearly unassociated with proxy width alone.Width correlations are 0.034 for W and 0.017 for Z; mean relevance has a modest width correlation of ρ = 0.166 for both proxies.Source: Table B.7 · p. 40, §B.3.4 · p. 40
• Both proxies have median mean-column relevance of ≈0.28, but weakest-column relevance has median ≈0.05, with roughly 60% of datasets below the 0.1 weak line.More than 93% of datasets show positive typical-draw relevance; relevance rises with dimension slack and falls as proxy-mechanism noise share grows.Source: §B.3.4 · p. 40, Figure B.17 · p. 41

C.1 Architecture

• Queries contain only x⋆: adding proxy values would condition the prediction on proxies and change the target away from the CATE Q(x⋆).The query outcome and treatment are also absent because the estimand averages over both arms and is not realized for a unit in the table.Source: §C.1 · p. 42
• Structural role embeddings identify X, Z, and W because their causal roles cannot be inferred from feature values or fixed column positions.The treatment A is encoded with the outcome for context units, using separate projections selected by the treatment indicator.Source: §C.1 · p. 42
• A single CATE head predicts a distribution over query effects directly, avoiding an independence assumption between the posterior distributions of the two potential-outcome means.The output is a histogram over a fixed grid; the model reads its probabilities as a piecewise-uniform density.Source: §C.1 · p. 43
• The output grid is a hard support constraint, and its bin width sets a floor on any posterior standard deviation the head can report.The bounds must be wide enough to cover the CATE distribution produced by the prior.Source: §C.1 · p. 43

C.2 The noise-aware histogram loss

• The training objective scores a noise-convolved predictive density so the Monte Carlo variance cancels, allowing the learned distribution to recover the posterior as smoothing is annealed.Directly fitting noisy Monte Carlo targets instead would add their variance to the learned posterior, with greater inflation where treatment-effect heterogeneity is larger.Source: §C.2 · p. 45

• After the smoothing anneal, the distributional loss and total objective continued to decrease through step 75,000, while the mean term remained a small regularizer.The mean term followed the distributional term closely.Source: §C.3 · p. 46
• Only 0.02% of synthetic-prior CATE targets fell outside the output grid on average, although the out-of-grid share exceeded 1% at 275 of 75,000 steps.Targets outside the grid cannot be represented by the output head.Source: §C.3 · p. 46
• The mean realized Monte Carlo replicate count was 250—the requested value—at every logged step, so the generator’s memory cap never bound.This preserved the full replicate budget for the CATE targets.Source: §C.3 · p. 46
• The single training run consumed 66.15 H100 GPU-hours, with synthetic-prior generation accounting for 43.46 GPU-hours, or 65.7%.The run used 75,000 steps and 4.8 million synthetic datasets.Source: §C.3 · pp. 45, 47

C.3.1 Post-training on larger context sizes

• The post-training experiment expanded context sizes from the main run’s maximum of 2,048 to a uniform range of 64–5,000 observations.It ran for 30,000 steps on the same causal prior and noise-aware objective, with 128 query rows.Source: §C.3.1 · p. 47
• To limit changes to the learned representation, adaptation froze all parameters except the in-context predictor, including its first six transformer blocks.Checkpoints were evaluated on separate frozen prior and realistic validation suites at large context sizes.Source: §C.3.1 · p. 47
• The training-configuration table documents the prior, objective and smoothing schedule, optimizer, initialized backbone, and checkpoint-selection validation sets.Source: Table C.2 · p. 48

C.4.1 Validation datasets

• The fixed validation suite contains 480 datasets balanced across 12 realistic mechanisms, spanning diverse variable types, outcomes, and confounding settings.Each episode has 2,048 context rows and 128 query points, with target CATEs estimated from 10,000 Monte Carlo replicates.Source: §C.4.1 · p. 49

C.4.2 Point-estimate validation

• Point-estimate evaluation covers pooled ATE RMSE and normalized bias, plus CATE nPEHE and shape R2.Shape R2 isolates treatment-effect heterogeneity recovery from constant level error.Source: §C.4.2 · p. 49
• After about 3,500 steps, median CATE nPEHE beat the constant-ATE baseline and shape R2 became positive; point-estimate metrics then continued improving through training.ATE RMSE decreased, normalized ATE bias moved toward zero, and the interquartile bands narrowed across episodes.Source: §C.4.2 · p. 49, Figure C.2 · p. 50

C.4.3 Distributional validation

• CRPS decreased during training, but predictive spread and empirical coverage converged to overconfident states.The figure tracks CRPS, predictive standard deviation relative to RMSE, empirical coverage, and credible-interval widths.Source: §C.4.3 · p. 51, Figure C.3 · p. 51

C.4.4 Posterior representation & smoothing

• Savitzky–Golay smoothing of the CDF makes posterior densities easier to interpret without changing the distributions used for validation metrics.The authors identify training-time regularization for smoother posterior estimates as future work.Source: §C.4.4 · pp. 51–52, Figure C.4 · p. 52

C.4.5 Posterior dynamics

• At early checkpoints, posterior widths remain similarly wide across context sizes, and the fixed-query posterior is diffuse and weakly context-dependent.Source: §C.4.5 · p. 52, Figure C.5 · p. 53, Figure C.6 · p. 53
• At the final checkpoint, mean predictive standard deviation falls from approximately 0.54 at nctx = 16 to 0.10 at nctx = 2 048.Later training produces context-dependent contraction, with larger contexts yielding narrower posteriors.Source: §C.4.5 · p. 52, Figure C.5 · p. 53
• For the fixed query point, training concentrates the posterior around the true CATE, and the larger context produces a noticeably sharper final posterior.Source: §C.4.5 · p. 52, Figure C.6 · p. 53

Appendix D.1.5

• The catalogue includes proximal KPV and PMMR bridges with either empirical-marginal W integration or locally conditional W | X = x integration.Source: Table D.1 · pp. 54–55, Appendix D.1.2 · p. 54, Appendix D.1.5 · p. 54

D.1.1 Backdoor meta-learners, in-context learners and proxy marginalization

• Backdoor baselines include S-, T-, and X-learners and zero-shot in-context learners, but hidden U violates conditional ignorability, so they serve as empirical baselines.The evaluated in-context methods are CausalPFN, CausalFM, and Do-PFN.Source: §D.1.1 · pp. 55–56
• Proxy-conditioned predictions are averaged over aligned (W, Z) neighbors using k-NN or Gaussian weights; even this marginalization does not identify Q(x) by naive regression.The benchmarks use k = 16 and kernel scale 1.0.Source: §D.1.1 · pp. 55–56

D.1.2 P-learner

• The P-learner estimates cross-fitted doubly robust proximal scores using outcome and treatment bridges, then regresses those scores on X to obtain the CATE.Its bridges use regularized minimax RKHS estimation with Nyström features and cross-validated penalties.Source: §D.1.2 · p. 57

D.1.3 Neural moment matching regression

• NMMR fits a neural outcome bridge and adapts its ATE-oriented reference implementation to CATE prediction by marginalizing W at each query point.It evaluates marginal-W integration under W ⊥⊥ X and conditional W | X = x integration, the latter approximating the proximal g-formula without that independence assumption but requiring extra smoothing.Source: §D.1.3 · pp. 57–58
• NMMR reports V- and U-statistic variants, removes an output ReLU that prevented negative predictions, and evaluates median-heuristic bandwidths alongside the fixed bandwidth.The fixed bandwidth can degenerate in higher dimensions, losing proxy information in the V-statistic and causing instability in the U-statistic.Source: §D.1.3 · p. 58

D.1.4 Proxy maximum moment restriction

• PMMR uses an RKHS outcome bridge with a closed-form minimizer and adapts its prediction heads to the same independent- or conditional-W CATE strategies as NMMR.For conditional W | X = x integration, PMMR uses model kernel weights and requires no extra hyperparameters.Source: §D.1.4 · p. 59
• For binary treatment, PMMR is solved arm by arm exactly, reducing computational and memory costs; bandwidths and regularisers are selected per episode using the held-out V-statistic.The settings sweep regulariser combinations for each sampled bandwidth draw.Source: §D.1.4 · p. 60, Table D.4 · p. 60

D.1.5 Kernel proxy variable

• KPV estimates the same outcome bridge as PMMR in the same RKHS, using two ridge regressions: first a conditional mean embedding of W, then outcome regression.Following the released implementation, both stages reuse all context rows; KPV explicitly estimates W conditional on A, X, and Z.Source: §D.1.5 · p. 60
• KPV selects both regularisers by closed-form leave-one-out criteria within each fit, and tunes a shared bandwidth multiplier per episode using stage-2 LOOCV loss.Bandwidth search is initialized by the median heuristic; this uses outcome prediction error rather than the stage-1 criterion used in the released implementation.Source: §D.1.5 · p. 60, Table D.5 · p. 61
• For CATE prediction, KPV uses the same two strategies as PMMR and marginalizes over joint (X, W), unlike released code’s separate Gram averaging, which agrees only if they are sample-independent.The reference averages the fitted bridge on a separate test sample; this implementation uses context rows for both fitting and averaging.Source: §D.1.5 · p. 60
• KPV corrections reuse context-estimated bandwidths and scaling consistently at prediction, and its binary treatment kernel permits arm-by-arm closed-form computation without changing the estimator.The original implementation recomputed median bandwidths per kernel call, used a default prediction scale, and separately scaled test inputs.Source: §D.1.5 · p. 61

D.2 Semi-synthetic benchmarks from real covariates

• Twelve INRIA-SODA datasets yield 96 configurations crossing linear or nonlinear mechanisms, weak or strong confounding, and weak or strong proxy information, with three realizations each.Real source rows provide the joint distribution of observed and hidden variables; generated mechanisms define treatment, proxies, and outcomes.Source: §D.2 · p. 61, Figure D.1 · p. 62
• Sources have at least 20 features and 10 000 rows; fixed calibration rows set preprocessing, while evaluation episodes sample context and query rows from the remaining pool.Retained source columns are assigned equally to observed X and hidden source variables, with d = 5 in the experiments.Source: §D.2 · p. 61
• Latent U combines source features with independent Gaussian innovation, retaining dependence with X while adding variation at fixed X.A calibration transformation and covariance normalization produce a unit-standard-deviation latent score shared by treatment and baseline outcome.Source: §D.2 · p. 61
• Observed mechanisms use linear X coordinates or nonlinear transformed coordinates and pairwise products, with standardized functions defining treatment, outcome, CATE, and proxy offsets.The CATE function Q(X) has calibration mean 0.5 and standard deviation one.Source: §D.2 · p. 62
• Proxy measurements combine U-dependent channels, observed-X offsets, and independent Gaussian errors; r = 0.2 and r = 0.8 define weak and strong proxy settings.The measurement map is identity for linear mechanisms and componentwise asinh for nonlinear mechanisms; joint information about U also depends on the source distribution and both channels.Source: §D.2 · p. 62
• Shared potential-outcome disturbances make the exact heterogeneous effect Q(X); the construction also supplies an explicit outcome bridge under its stated conditional-independence relations.For nonlinear mechanisms, the bridge is retained through invertible measurement transformations.Source: §D.2 · p. 63
• Configurations share source-row assignments, and within mechanism families they also share fitted functions and disturbances; realizations independently redraw rows, mechanisms, and disturbances.Different realizations may use overlapping sampled rows.Source: §D.2 · p. 63

D.3 Results on semi-synthetic datasets

• Post-training moves mainly toward lower squared bias in the query-average treatment effect, with little improvement in CATE shape in the nonlinear, high-confounding, low-proxy-reliability setting.ProximalFM-FT continues training the main checkpoint on the same prior and objective, using contexts up to 5 000 rows.Source: Figure D.2 · p. 64
• On semi-synthetic datasets, credible intervals increasingly undercover as context grows even though predicted posterior uncertainty contracts.The prior-drawn evaluation intervals stay close to nominal coverage.Source: Figure D.3 · p. 65

D.3.1 Speed

• Runtime comparisons use sequential execution on 36 nonlinear, strongly confounded, weak-proxy episodes and measure construction, fitting, and prediction at each context size.The passage describes timing methodology but supplies no runtime outcomes.Source: §D.3.1 · p. 71

Joint proxy information

• For each dimension, the study generates 20 proxy-information levels from 0.01 to 0.99 and uses joint population R2 for predicting U from (W, Z) as the information measure.The joint R2 equals s, and conditioning additionally on X preserves it because X is independent.Source: §D.4 · p. 71

Causal mechanism and heterogeneity

• The treatment effect is Q(X) = 0.5 + SX, while individual effects also vary with latent U.Across dimensions, normalizing by √d keeps the population scales of confounding and heterogeneity fixed.Source: §D.4 · p. 71

Joint proxy information

• The true outcome bridge remains valid as proxy information falls, but proxy-score error variance (1 − q)/q quantifies increasing noise amplification.The bridge contrast averaged over W | X recovers Q(X).Source: §D.4 · p. 72

Estimator comparison

• The study compares ProximalFM with interacted proximal 2SLS, unpenalised OLS, and proxy-adjusted CausalPFN using shared training rows and 256 held-out query rows.CausalPFN treatment contrasts are marginalized using 16 seeded, aligned proxy pairs sampled from context.Source: §D.4 · p. 72

Results

• At nctx = 2 048, ProximalFM outperforms proxy-marginalized CausalPFN throughout the proxy-information sweep and proxy-adjusted OLS in most settings.As proxies nearly reveal U, OLS catches up with and surpasses ProximalFM; the curves use one paired simulation per setting.Source: §D.4 · p. 72, Figure D.4 · p. 73

Physical setup and causal structure

• The experiment combines physical responses and software controls to link hidden confounding, proxies, treatment, and outcome in a non-simulated optical system.It is presented as an exploratory step toward proximal estimation in real-world systems.Source: E Towards a real physical system:… · p. 74, Figure E.2 · p. 77
• The confounder U is an LED drive, X controls a red LED, Z the ultraviolet LED 2, W the first polariser position, A the ultraviolet LED 3 drive, and Y an infrared sensor reading.The setup also records electrical current and additional infrared and visible-light readings.Source: E Towards a real physical system:… · p. 74

ATE reference and interpretation

• The physical estimand is the ATE over acquired operating states, referenced by applying drive zero and then drive a1 after measuring Y.Its causal interpretation assumes stable device state and negligible systematic measurement-order or carryover effects.Source: E Towards a real physical system:… · p. 75
• At drive 500, mean intervention contrasts are approximately 886–893 sensor counts, versus 621–624 counts at drive 350.Across conditions, individual-contrast standard deviation is 3.55–65.64 counts, and the root mean squared difference from a remeasured factual arm is 3.10–65.00 counts.Source: E Towards a real physical system:… · p. 75

Adaptation of methods for ATE estimation

• Methods are fitted only on factual context rows; paired intervention measurements are reserved for the reference ATE.Conditional-effect predictions are averaged over the same context prefix.Source: E Towards a real physical system:… · p. 76

Results

• No method is best in every physical setting; X-learner is most consistently accurate, while proximal methods, including ProximalFM, perform well only in some settings.Larger contexts do not lead to systematic improvements, and CausalPFN is also competitive for nonlinear mechanisms and weak confounding.Source: E Towards a real physical system:… · p. 79, Figure E.4 · p. 80