ProximalFM: Amortized Proximal Causal Inference under Hidden Confounding

Christophe Muller; Ayub Kharel; Alex Luedtke; Chan Park; Eric Tchetgen Tchetgen; Juan L. Gamella; Rahul Krishnan; Ricardo Silva

Figure B.1: Detailed diagram of the proximal prior’s per-dataset generation pipeline.

hyperparameter	value	description
Block dimensions
dim_X	$\mathrm{Unif}\{0,\dots,\texttt{max\_dim\_X}\}$	observed baseline covariate dimension
dim_U	triangular on $\{1,\dots,\texttt{max\_dim\_U}\}$	latent confounder dimension
dim_W	$\mathrm{Unif}\{d_{U},\dots,\texttt{max\_dim\_W}\}$	outcome-inducing proxy dimension
dim_Z	$\mathrm{Unif}\{d_{U},\dots,\texttt{max\_dim\_Z}\}$	treatment-inducing proxy dimension
Dimension bounds

Appendix

Table B.1: Structural parameters of the prior: the block dimensions and the dataset shape. Each dimension is drawn per dataset from the law shown, stated in terms of the ceilings listed below it. The proxy dimensions are drawn conditionally on

d_{U}

so that the completeness heuristic of Section B.2.2 .

hyperparameter	value	description
Mechanism
noise_std	$0.001$	tiny fixed per-layer dither so continuous columns realise distinct values
Feature corruption
max_categories	$50$	cap on categories per feature
proxy_cat_cover_prob	$0.5$	chance a proxy mirrors a $U$ category (vs. a continuous wildcard)
Positivity

Appendix

Table B.2: Fixed hyperparameters of the prior, held constant across every dataset. Listed are the constants that pin down behaviour the surrounding text describes only qualitatively; switches whose shipped value is that behaviour itself are stated in the text instead.

hyperparameter	sampling law	description
Mechanism
num_layers	log-scaled trunc. normal, mean $\sim\log\mathrm{Unif}[1,6]$ , int $\geq 2$	mechanism MLP depth
hidden_dim	log-scaled trunc. normal, mean $\sim\log\mathrm{Unif}[5,130]$ , int $\geq 4$	mechanism MLP width
mlp_activations	categorical over 54 random activation functions	activation, drawn from a random library
init_std	log-scaled trunc. normal, mean $\sim\log\mathrm{Unif}[0.01,10]$ , real $\geq 0$	weight-initialisation scale
block_wise_dropout	categorical $\{True,False\}$ (random weights)	block-sparse vs. dense weight init

Appendix

Table B.3: Per-dataset sampled hyperparameters of the prior. Each is drawn once per dataset from the listed meta-distribution; log-scaled truncated normals draw their centre log-uniformly over the stated mean range and add the lower bound as an offset.

Figure B.2: Delivered marginal distributions of the block dimensions

d_{U},d_{W},d_{Z}

across the prior, for the shipped (triangular

d_{U}

) and the alternative (uniform

d_{U}

) sampling, overlaid with their analytic sampling laws (dashed: triangular; dotted: uniform). Bars are empirical frequencies over

1024

datasets. Uniform

d_{U}

pushes proxy mass against the ceiling

d^{\max}=10

; the triangular law spreads it into the linear ramp of ( 24 ). Delivered bars track the theoretical curves, confirming the generator samples the intended laws.

Figure B.3: Every dial: each input channel’s share of

f_{A}

’s (top) and

f_{Y}

’s (bottom) total input variance , against its own sampled knob (

s

for the

U

-channel, the covariate and proxy keep-probabilities for

\tilde{X}

and

\tilde{Z}_{A}/\tilde{W}_{Y}

, the noise share for

\varepsilon

). Solid line: moving average. Each knob steers its channel’s delivered variance share.

Figure B.4: Output side of the composition: what each mechanism’s output depends on , measured by a cross-fitted TabICL regression on the delivered data. Columns 1–3 plot each parent channel’s unique leave-one-out

\Delta R^{2}

(drop in out-of-fold

R^{2}

when that block is removed from the predictors) against its own knob, for

f_{A}

(top, target

\mathrm{logit}(\pi)

) and

f_{Y}

(bottom, target

Y_{0}

); column 4 plots the unexplained share

1-R^{2}

against the noise knob. Solid line: moving average. Each channel’s dependence rises with its knob, as at the input level. The

U

-channel reads modestly because leave-one-out credits only signal unique to

U

, not the part shared with its child proxy.

Proxy profile $P$	Heuristic?	Reason
$[\,\mathrm{num},5\text{-cat},3\text{-cat}\,]$	ok	exact match, column for column
$[\,\mathrm{num},8\text{-cat},4\text{-cat}\,]$	ok	categorical columns carry more categories than required
$[\,\mathrm{num},\mathrm{num},3\text{-cat}\,]$	ok	a numerical column covers $U$ ’s $5$ -cat column
$[\,\mathrm{num},\mathrm{num},\mathrm{num}\,]$	ok	numerical columns dominate every requirement
$[\,\mathrm{num},\mathrm{num}\,]$	no	too few columns ( $d_{P}=3<d_{U}=4$ )
$[\,\mathrm{num},4\text{-cat},3\text{-cat}\,]$	no	the $4$ -cat column cannot cover $U$ ’s $5$ -cat column

Appendix

Table B.4: Worked examples of the categorical-completeness heuristic for the fixed confounder profile

U=[\,\mathrm{num},\,5\text{-cat},\,3\text{-cat}\,]

(one numerical column and two categorical columns, with

5

and

3

categories). A proxy block

P

satisfies the heuristic when its columns can be matched one-to-one to those of

U

so that each is at least as informative: a numerical column dominates anything, and a categorical column dominates a categorical requirement only if it has at least as many categories. A numerical proxy column may thus cover a categorical requirement, but a categorical proxy column can never cover a numerical one.

Figure B.5: Joint distribution of the latent dimension

d_{U}

against each feature block (

d_{X}

,

d_{W}

,

d_{Z}

), one row per prior variant ( triangular , uniform ). Colour encodes the number of delivered datasets falling in each

(d_{U},\cdot)

cell. The diagonal line on the

d_{W}

/

d_{Z}

panels marks the completeness heuristic

d_{W},d_{Z}\geq d_{U}

; both variants place their mass on or below it, i.e. constant-column filtering does not measurably break the heuristic imposed at sampling time.

d_{X}

carries no such constraint and is shown for reference.

Figure B.6: Column-type composition of each block (

U

,

W

,

Z

), one

100\%

-stacked bar per variant ( default : categorical completeness enforced; no_enforcement : proxies left unconstrained). Each bar partitions a block’s columns into continuous (grey, “cont.”) and categorical, the latter binned by category count (

2

–

5

singletons, then coarser

6

–

10

,

11

–

20

,

21

–

d^{\max}_{\mathrm{cat}}

tail bins). Continuous columns dominate every block; contrasting

W

/

Z

across the two variants shows how much enforcement pushes proxy columns categorical relative to the unconstrained no_enforcement arm.

Figure B.7: How enforcement makes proxy structure track

U

’s categorical demand, pooling

W

and

Z

across the default and no_enforcement variants; shaded bands are

\pm

one SEM. (a, b) condition on a single

U

column’s own cardinality: the

x

-axis bins a

U

column by its category count (continuous “ num ”, then

2,\dots,9

, and coarser tail bins up to “

30{+}

”), and each point averages a statistic of the co-occurring proxy block over every dataset containing a

U

column of that kind – (a) the fraction of proxy columns that are continuous, essentially flat in

U

’s cardinality but higher under default , and (b) the mean category count of the proxy’s categorical columns, which rises with

U

’s cardinality under default but stays low under no_enforcement . (c) moves to the dataset level: the mean fraction of categorical proxy columns against

U

’s total categorical demand

|\mathrm{cats}(U)|

, with bins of fewer than

5

datasets dropped. Both arms rise (the per-dataset categorisation propensity is shared across blocks) and default sits below no_enforcement : enforcement deepens the proxy’s categorical columns rather than multiplying them. Because proxy statistics in (a, b) are block-level rather than tied to a matched column, those trends reflect association with “the dataset contains a

U

column of this kind,” not a one-to-one mapping.

Figure B.8: The treatment-assignment pipeline of ( 25 )–( 29 ) on one representative dataset, tracking a single unit (red line) across all five stages. (a) The raw logit

\ell=f_{A}(\cdot)

, whose scale and offset are an uncontrolled by-product of the sampled architecture. (b) The standardised

\tilde{\ell}

of ( 26 ), pinned to zero mean and unit variance. (c) The affine

\ell^{\prime}=s\,\tilde{\ell}+b

of ( 27 ), shifted and sharpened by the two sampled overlap knobs. (d) The propensity

\pi=\sigma(\ell^{\prime})

of ( 28 ), confined to

[\epsilon,1-\epsilon]

by the clamp of ( 29 ) (dotted bounds, drawn at an enlarged

\epsilon

for visibility; production

\epsilon=10^{-3}

). (e) The realised treatment

A\sim\mathrm{Bernoulli}(\pi)

across the batch (fraction in each arm), with the tracked unit’s realised arm marked in red.

Figure B.9: The heterogeneity dial on the delivered prior, over

1024

datasets. All statistics are ratios, since each SCM’s outcome carries its own arbitrary scale. (a) Two datasets’ effect densities on a shared outcome-scaled axis, each centred on its own

\bar{\tau}

(dashed line), one gathered (

\gamma=0.09

) and one spread (

\gamma=0.91

); the pair is chosen with matched raw dispersion

c=\mathrm{sd}(\tau^{\mathrm{raw}})/\mathrm{sd}(Y)

(

1.127

vs.

1.125

), so the contrast is attributable to

\gamma

rather than to SCM variation. (b) The outcome coupling

\mathrm{corr}(Y_{0},Y_{1})

against

\gamma

(solid line: moving average). At

\gamma=0

the effect is the constant

\bar{\tau}

, so

Y_{1}=Y_{0}+\bar{\tau}

and the coupling is exactly

1

(dashed reference). (c) Distribution of the delivered dispersion

\mathrm{sd}(\tau)/\mathrm{sd}(Y)

per

\gamma

bin (Spearman

0.78

overall), against the proportionality

\gamma\cdot\mathrm{median}(c)

implied by the collapse; within-bin spread is dominated by variation in

c

. Because the reference line is anchored at zero and uses the overall median

c

for its slope, the panel verifies strict proportionality to

\gamma

—linear scaling starting from the origin—rather than predicting the absolute magnitude of dispersion.

Figure B.10: Gallery of feature marginals: one column per block (

X

,

U

,

W

,

Z

, the propensity

\pi=P(A{=}1\mid\cdot)

for

A

, both potential outcomes for

Y

), one row per randomly sampled dataset. Each panel is one feature column’s histogram, solid for continuous columns and dashed for categorical ones; the

Y

column overlays

Y_{0}

(blue) and

Y_{1}

(red). Feature blocks share the standardised axis

\mathrm{Corrupt}

leaves;

\pi\in[0,1]

and

Y

keep their natural scale.

Figure B.11: Within-block dependence: one column per block, one randomly sampled dataset per row, each panel a scatter of two features from that block. For

X

,

U

,

W

,

Z

the two features are random columns of the block (sampled from datasets of width

\geq 2

); for

A

the panel plots the propensity

\pi

against the treatment draw

A

; for

Y

the two potential outcomes

Y_{0}

vs

Y_{1}

. Rows are subsampled for legibility; feature axes inherit the standardised scale.

Figure B.12: Cross-block dependence: one column per chosen block pair, one sampled dataset per row, each panel a scatter of one feature from each block.

U

–

W

and

U

–

Z

are the parent

\to

child edges;

W

–

Z

is the proxy pair, associated only through the shared latent

U

(no direct edge);

U

–

\pi

is the confounder against the propensity

\pi=P(A{=}1\mid\cdot)

. Rows are subsampled for legibility; feature axes inherit the standardised scale.

Figure B.13: Delivered overlap across the prior, from the stored true propensity

\pi=\Pr[A=1\mid X,U]

(no estimation involved). (a) Pooled distribution of the overlap margin

\min(\pi,1-\pi)

over all rows of

1024

datasets, on a logarithmic axis: the margin measures each unit’s distance from a degenerate assignment, folding both tails into one axis, and the log scale is what makes the mass piled on the clamp

\varepsilon=10^{-3}

(dotted) visible at all. Bars are per-bin shares of all rows on a logarithmic grid, so bin width grows to the right. (b) Per-dataset quantile function of

\pi

, summarised across datasets by the median curve with the

25

–

75\%

and

10

–

90\%

inter-dataset bands. Dashed lines mark the practical-overlap band

[0.05,0.95]

in both panels.

Figure B.14: Exclusion-restriction audit ( Assumption 2 ) over

1024

datasets. Each panel is one clause – (a)

Z

excluded from

f_{Y}

, (b)

W

excluded from

f_{A}

– and each column a cross-fitted incremental

\Delta R^{2}

of the excluded block: conditioning on

U

is the check (expected

\approx 0

), control:

U

dropped the positive control (expected clearly positive). Colour and horizontal position both encode

u\_dependence=R^{2}(\mathrm{with}\ U)-R^{2}(\mathrm{without}\ U)

, i.e. how much confounding that draw carries (linear ramp, floored at

0

, capped at the

95

th percentile). Both checks stay on zero regardless of colour; both controls fan upward with warmer colour, most visibly in the top tercile of

u\_dependence

( Table B.5 ).

conditioning set	role	datasets	median $\Delta R^{2}$	90th pct	share $>0.02$
$Z$ excluded from $f_{Y}$
$\{A,X,U,Z\}$	conditions on $U$	all	-0.002	0.007	0.021
$\{A,X,U,Z\}$	conditions on $U$	top confounded	-0.005	0.004	0.012
$\{A,X,Z\}$	positive control	all	0.004	0.099	0.313
$\{A,X,Z\}$	positive control	top confounded	0.029	0.222	0.598
$W$ excluded from $f_{A}$

Appendix

Table B.5: Exclusion-restriction audit ( Assumption 2 ): cross-fitted incremental

\Delta R^{2}

of the excluded block over the stated conditioning set, for the check (conditions on

U

) and the positive control (

U

dropped). Top confounded is the third of datasets with the largest

u\_dependence

(

R^{2}

with

U

minus

R^{2}

without it).

Figure B.15: Confounding-strength comparison ( Table B.6 ) over

1024

datasets. Each violin shows normalised absolute ATE error

|\widehat{\mathrm{ATE}}-\mathrm{ATE}|/\mathrm{sd}(Y_{\mathrm{obs}})

on a log axis, scored against

\mathrm{mean}(Y_{1}-Y_{0})

: naive (no adjustment), backdoor

X

and

(X,W,Z)

(T-learners on measured covariates), and oracle

(X,U)

and

(X,U,W)

(T-learners also given the latent confounder).

Figure B.16: Backdoor-

X

and oracle-

(X,U)

error over the delivered confounding-share grid:

1024

datasets binned

3\times 3

on the quantiles of the delivered

U

share of

f_{A}

(rows) and

f_{Y}

(columns) ( Figure B.3 ). Cells report the median normalised error on a shared colour scale. The backdoor panel’s top row (delivered

U

share of

f_{A}

above

0.495

) is elevated at

0.045

–

0.046

against

0.028

–

0.041

in the two rows below, and is nearly flat across

f_{Y}

’s share. The oracle panel shows no comparable pattern.

rung	adjustment set	role	median error	90th pct	share $>0.1$
naive	none	unadjusted	0.041	0.195	0.241
backdoor X	$X$	correct if no latent confounder	0.037	0.163	0.211
backdoor XWZ	$(X,W,Z)$	DAG-blind: everything measured	0.030	0.121	0.148
oracle XU	$(X,U)$	identification: minimal sufficient set	0.028	0.104	0.108
oracle XUW	$(X,U,W)$	efficiency: adds $W$ , a parent of $f_{Y}$	0.025	0.105	0.107

Appendix

Table B.6: ATE error across adjustment sets. Errors are absolute deviations from the in-sample

\mathrm{mean}(Y_{1}-Y_{0})

, divided by

\mathrm{sd}(Y_{\mathrm{obs}})

. All adjusted estimates use cross-fitted T-learners; the oracle rows include the latent

U

.

Figure B.17: Proxy relevance measured by the cross-fitted increment

\Delta R^{2}_{P}(u_{j})=R^{2}(u_{j}\mid X,P)-R^{2}(u_{j}\mid X)

, shown separately for

W

and

Z

. (a) Distribution of the mean and weakest-column

\Delta R^{2}

over

U

; boxes show the IQR and whiskers the

10

th–

90

th percentiles. (b) Median

\Delta R^{2}

by proxy dimension slack

d_{P}-d_{U}

, with slack

\geq 7

pooled. (c) Median

\Delta R^{2}

by sampled noise share of

f_{P}

, in six equal-frequency bins per proxy. Bands in (b) and (c) show the IQR.

$W$
aggregation over $U$	median $\Delta R^{2}$	share $<0.1$	$\rho$ $d_{P}$	$\rho$ $d_{U}$	$\rho$ $d_{P}-d_{U}$	$\rho$ noise
mean	0.273	0.226	0.166	-0.002	0.208	-0.202
weakest	0.053	0.593	0.034	-0.249	0.277	-0.140
$Z$
mean	0.271	0.235	0.166	-0.023	0.226	-0.208
weakest	0.052	0.588	0.017	-0.272	0.281	-0.095

Appendix

Table B.7: Proxy-relevance audit ( Assumption 4 ): cross-fitted incremental

\Delta R^{2}=R^{2}(u_{j}\mid X,P)-R^{2}(u_{j}\mid X)

of each proxy

P

for the latent columns, aggregated over

U

by the mean or by the weakest column. The counting heuristic ( Section B.2.4 ) passes for every dataset, so the share column is the rate at which it and functional recoverability disagree.

\rho

is Spearman’s correlation with each candidate driver: the proxy’s own width

d_{P}

,

d_{U}

, the slack

d_{P}-d_{U}

that conflates the two, and the sampled noise share of the

f_{P}

input.

Stage	Hyperparameter	Value
Shared	Embedding dimension $E$	128
	Feedforward expansion factor	2
	Activation	GELU
	Normalization	pre-norm
	Dropout	$0.0$
Column embedder	Blocks	3

Appendix

Table C.1: Architectural hyperparameters of ProximalFM . The three stages are those of TabICLv2, kept at its values so that its pre-trained weights initialize every component we did not modify; the output head is ours. Parameter counts are per stage, the in-context learner’s including the head.

Stage	Hyperparameter	Value
Prior	Confounder dimension $d_{U}$	$1$ – $10$
	Treatment proxy dimension $d_{Z}$	$1$ – $10$
	Outcome proxy dimension $d_{W}$	$1$ – $10$
	Covariate dimension $d_{X}$	$0$ – $10$
	Context units $N$	$64$ – $2\,048$ (uniform)
	Query units $n_{q}$	128

Appendix

Table C.2: Training configuration of ProximalFM : the prior the model is trained on, the objective and its smoothing schedule, the optimizer, the pre-trained backbone it is initialized from, and the validation sets used for checkpoint selection.

Figure C.1: Training curves for ProximalFM . (a) The objective and its two terms against log step, with the

\sigma

-annealing window shaded; the dashed line is the loss of a uniform density over the CATE grid, matching the value at initialisation. (b) The same curves on a linear step axis after the anneal. (c) The

\sigma

and learning-rate schedules. (d) A magnified view of the weighted mean term,

\lambda_{\mathrm{mean}}\mathcal{L}_{\mathrm{mse}}

, with the

\sigma

-annealing window again shaded. Stochastic curves show the median and interquartile band within log-spaced step bins in panels (a) and (d), and linearly spaced bins in panel (b); the deterministic schedules in panel (c) are shown without binning.

		Hours	%
Compute	Prior generation	$43.46$	$65.7$
	Forward / backward	$18.12$	$27.4$
	Validation, prior set (376 passes)	$1.27$	$1.9$
	Validation, realistic set (751 passes)	$2.34$	$3.5$
	Checkpointing, logging, startup	$0.96$	$1.5$
Total	GPU-hours ( $1\times$ H100 NVL)	$66.15$	$100.0$

Appendix

Table C.3: Cost of the ProximalFM training run, as measured GPU-hours. Validation is the two frozen evaluation datasets, scored at their logging intervals throughout training.

Figure C.2: Evolution of point-estimate performance during training on the fixed realistic-mechanism validation suite ( Section C.4.1 ). Curves show the validation metric at each evaluation step. The shaded regions in panels (b)–(d) indicate the interquartile range across validation episodes. (a) ATE RMSE, a pooled metric and therefore shown without an episode-level band. (b) normalized ATE bias, where zero denotes no bias. (c) CATE nPEHE, with the dashed line marking the constant-ATE baseline (

\mathrm{nPEHE}=1

). (d) CATE shape

R^{2}

, with the dashed line marking the corresponding constant-ATE baseline (

R^{2}_{\mathrm{shape}}=0

). Lower values are preferred in panels (a) and (c), while panel (b) is centered at zero and higher values are preferred in panel (d). For readability, curves and shaded bands are lightly smoothed by taking medians within log-spaced training-step bins.

Figure C.3: Evolution of distributional predictive performance during training on the fixed realistic-mechanism validation suite ( Section C.4.1 ). (a) CRPS, for which lower values indicate better probabilistic predictions. (b) predictive standard deviation divided by RMSE, with the dashed line at one denoting calibrated predictive spread. (c) empirical 90% and 50% coverages, with the dashed lines marking the nominal coverage levels and the shaded regions showing the interquartile ranges across validation episodes. (d) widths of the central 90% and 50% credible intervals, pooled over validation rows. Panels (b) and (d) use logarithmic y-axes. For readability, curves are lightly smoothed by taking medians within log-spaced training-step bins.

Figure C.4: Illustration of the spiky posterior-density representation for one selected query point from the validation suite. (a) Raw posterior density induced by the model’s histogram output. (b) Corresponding raw CDF. (c) CDF after Savitzky–Golay smoothing with a 50-point window and polynomial degree three. (d) Density obtained by differentiating the smoothed CDF. The smoothed CDF closely follows the raw CDF while yielding a more visually interpretable density. The dashed line marks the true CATE. Smoothing is used for visualization only; all validation metrics are computed from the original posterior representation.

Figure C.5: Posterior-width contraction during training on the fixed realistic-mechanism validation suite ( Section C.4.1 ). Each cell reports the mean predictive posterior standard deviation averaged over all validation query rows, for a given training checkpoint and context size. Widths are expressed in standardized units and use the piecewise-uniform posterior variance. The shared colour scale is logarithmic.

Figure C.6: Evolution of the predictive CATE posterior for one fixed query point from the realistic-mechanism validation suite ( Section C.4.1 ). Curves show the density obtained from the posterior CDF at four training checkpoints, with training steps indicated by colour. The same query point is evaluated with (a)

n_{\mathrm{ctx}}=64

and (b)

n_{\mathrm{ctx}}=1\,024

. The dashed line marks the true CATE. Densities are shown in standardized units. The CDFs are smoothed for visualization, following Section C.4.4 .

Predictor	Family	Fitted estimator and prediction rule	Details
S-learner (ridge)	Backdoor meta-learner	S-learner with ridge regression.	Section D.1.1
S-learner (RF)	Backdoor meta-learner	S-learner with random forest.	Section D.1.1
S-learner (TabICL)	Backdoor meta-learner	S-learner with zero-shot TabICL.	Section D.1.1
S-learner (TabICL) + kNN marg.	Backdoor meta-learner	S-TabICL fit on $(X,W,Z)$ , then 16-neighbour proxy marginalisation.	Section D.1.1
S-learner (TabICL) + kernel marg.	Backdoor meta-learner	S-TabICL fit on $(X,W,Z)$ , then 16-neighbour kernel proxy marginalisation (scale 1).	Section D.1.1
T-learner (ridge)	Backdoor meta-learner	T-learner with ridge regression.	Section D.1.1

Appendix

Table D.1: Catalogue of CATE predictors selected in the main benchmark.

Component	Setting	Value
Cross-fitting	Folds (stratified on $A$ )	4
Bridge class	Kernel	RBF, blockwise median heuristic
	Nyström rank $r$	256
Bridge regularization	$\lambda_{h},\lambda_{q}$	sweep : $\{10^{-6},10^{-5},10^{-4},10^{-3},10^{-2},10^{-1}\}$
	Selection criterion	Held-out proximal moment loss
	Validation fraction	$0.25$

Appendix

Table D.2: P-learner settings. Bridge regularization parameters are selected separately within each cross-fitting fold using held-out moment loss. Final-stage hyperparameters are selected per episode using four-fold cross-validated prediction MSE.

Component	Setting	Value
Bridge class	MLP, ReLU, linear output
	Hidden width	searched : $\{16,32,64\}$
	Depth (hidden layers)	searched : $\{2,4,6\}$
Moment kernel	RBF over $(A,Z/\hat{\sigma}_{Z},X/\hat{\sigma}_{X})$
	Length scale $\ell$	fixed at $1.0$ or median heuristic
Loss	U- or V-statistic ( 50 )	one method each

Appendix

Table D.3: NMMR settings. Values marked searched are selected per episode by the procedure described below; the remainder are fixed across all episodes. The U- and V-statistic variants are reported as separate methods, so the loss is fixed within a run rather than searched.

Component	Setting	Value
Kernels	Bandwidths baseline	per-column median heuristic
	Kernel bandwidth multiplier (W,X)	searched : $\{0.8,1,1.2\}$
	Kernel bandwidth multiplier (Z)	searched : $\{0.8,1,1.2\}$
Regularisation	$\lambda$ (RKHS norm)	swept : $\{10^{-5},10^{-4},10^{-3},10^{-2},10^{-1}\}$
	Cholesky jitter	$10^{-8}$
Prediction	$W$ strategy	“ind. W”, “cond. W”

Appendix

Table D.4: PMMR settings. Bandwidth multipliers are sampled ; the regularisers are swept , i.e. every combination is evaluated for each bandwidth draw. Both are selected per episode on the held-out V-statistic.

Component	Setting	Value
Data split	$m_{1}$ (stage-1 rows)	$n_{\mathrm{context}}$
	$m_{2}$ (stage-2 rows)	$n_{\mathrm{context}}$
Kernels	Kernel family	Column-wise Gaussian RBF for $X,W,Z$ ; binary for $A$
	Base bandwidths	Per-column median heuristic, estimated on the context
	Shared multiplier $c$ on $\sigma_{d}^{2}$	Shared across $X,W,Z$ ; searched: $\{0.8,1.0,1.2\}$
Regularization	$\lambda_{1},\lambda_{2}$	searched by LOOCV; 10-point log grid in $[10^{-5},10]$

Appendix

Table D.5: KPV settings. Both stages use all context rows. The regularizers

\lambda_{1}

and

\lambda_{2}

are selected within each fit by their closed-form leave-one-out criteria. The shared kernel-scale multiplier is selected per episode using the stage-2 LOOCV loss.

Figure D.1: Construction of the generated proximal benchmarks. (a) A shared real source row supplies dependent observed covariates and hidden source features. Independent Gaussian innovation augments the hidden features. Both

X

and

U

enter the proxy measurements and the treatment and outcome mechanisms. (b) Independent Gaussian

X

and

U

define the controlled synthetic study. The joint proxy information parameter changes the measurement channels while preserving the causal variables within each dimension. The diagrams display the generated causal structure, with independent disturbances suppressed where indicated.

Source	Rows	Features
pol	$10\,082$	26
MiniBooNE	$72\,998$	50
default-of-credit-card-clients	$13\,272$	20
Higgs	$940\,160$	24
jannis	$57\,580$	54
heloc	$10\,000$	22

Appendix

Table D.6: Original sizes of the twelve selected datasets. Feature counts exclude the original prediction target, although that column remains eligible for selection as a source variable in our construction.

Figure D.2: Learning curves and error decomposition for the nonlinear, high-confounding, low-proxy-reliability setting (36 episodes: 12 sources with three realizations each). (a) Median CATE nPEHE versus context size; the dotted line marks the constant-ATE benchmark (

\mathrm{nPEHE}=1

). ProximalFM-FT continues training the main checkpoint on the same prior and objective with contexts up to

5\,000

rows, updating only part of the in-context predictor ( Section C.3.1 ). (b) For query-row errors

e_{i}=\hat{\tau}_{i}-\tau_{i}

, their mean

\bar{e}

, and true CATE variance

s_{\tau}^{2}=\operatorname{Var}_{i}(\tau_{i})

, the episode-level decomposition is

\mathrm{nPEHE}^{2}=\bar{e}^{2}/s_{\tau}^{2}+\operatorname{Var}_{i}(e_{i})/s_{\tau}^{2}

. Panel (b) plots the first term horizontally (squared bias in the query-average treatment effect) and the second vertically (heterogeneity error after removing that bias). Dotted contours show constant

\sqrt{x+y}

; plotted coordinates are separate component medians, so they need not reproduce panel (a)’s median nPEHE. Arrows and larger markers indicate increasing context size (

128

–

8\,192

). ProximalFM-FT moves mainly leftward: post-training improves the average effect estimate, with little improvement in CATE shape.

Figure D.3: Posterior calibration and contraction. The top row evaluates 400 datasets drawn from the pre-training prior; the bottom row evaluates 120 semi-synthetic datasets with nonlinear mechanisms, high confounding, and low proxy reliability. In every episode, the same

256

query points are evaluated as the context grows through nested prefixes from

128

to

8\,192

observations. (a,d) Coverage by context: each solid curve gives empirical coverage of a central CATE credible interval at one nominal level (1%, 5%, 10%, …, 95%, 99%); the faint dotted line of the same colour marks that level’s nominal coverage. (b,e) Calibration: each curve fixes a context size, coloured from purple (

128

) to yellow (

8\,192

), and plots empirical against nominal coverage. The dotted diagonal denotes perfect calibration; curves below it indicate undercoverage. (c,f) Posterior contraction: the black curve is the mean predicted CATE posterior standard deviation across query rows. The grey band is the interquartile range of episode-level mean standard deviations, showing variation between datasets. These panels use logarithmic vertical axes. Coverage stays close to nominal on the prior, whereas the semi-synthetic intervals increasingly undercover as context grows, even though their predicted uncertainty contracts.

	Linear				Nonlinear
	Low confounding		High confounding		Low confounding		High confounding
Method	Low proxy	High proxy	Low proxy	High proxy	Low proxy	High proxy	Low proxy	High proxy
S-learner (ridge)	1.027	1.027	1.458	1.458	1.059	1.059	1.459	1.459
S-learner (RF)	0.868	0.868	1.229	1.229	0.986	0.986	1.542	1.542
S-learner (TabICL)	0.782	0.782	1.336	1.336	0.937	0.937	1.457	1.457
S-learner (TabICL) + kNN marg.	0.885	0.895	1.070	0.921	0.973	0.982	1.259	0.986

Appendix

Table D.7: Exhaustive semisynthetic CATE nPEHE results at

n_{\mathrm{ctx}}=128

. Each entry first averages the three replicates of each source dataset, then averages equally over the 12 sources; lower is better. The lowest mean in each column is bold. Other underlined methods are not significantly worse than that empirical best under one-sided exact paired sign-flip tests across source-level means, with Holm correction within the column (

\alpha=0.05

).

	Linear				Nonlinear
	Low confounding		High confounding		Low confounding		High confounding
Method	Low proxy	High proxy	Low proxy	High proxy	Low proxy	High proxy	Low proxy	High proxy
S-learner (ridge)	1.013	1.013	1.564	1.564	1.023	1.023	1.580	1.580
S-learner (RF)	0.669	0.669	1.423	1.423	0.849	0.849	1.546	1.546
S-learner (TabICL)	0.499	0.499	1.294	1.294	0.762	0.762	1.432	1.432
S-learner (TabICL) + kNN marg.	0.527	0.581	1.053	0.683	0.793	0.822	1.206	0.865

Appendix

Table D.8: Exhaustive semisynthetic CATE nPEHE results at

n_{\mathrm{ctx}}=512

. Each entry first averages the three replicates of each source dataset, then averages equally over the 12 sources; lower is better. The lowest mean in each column is bold. Other underlined methods are not significantly worse than that empirical best under one-sided exact paired sign-flip tests across source-level means, with Holm correction within the column (

\alpha=0.05

).

	Linear				Nonlinear
	Low confounding		High confounding		Low confounding		High confounding
Method	Low proxy	High proxy	Low proxy	High proxy	Low proxy	High proxy	Low proxy	High proxy
S-learner (ridge)	1.011	1.011	1.567	1.567	1.024	1.024	1.593	1.593
S-learner (RF)	0.568	0.568	1.384	1.384	0.793	0.793	1.497	1.497
S-learner (TabICL)	0.379	0.379	1.256	1.256	0.653	0.653	1.393	1.393
S-learner (TabICL) + kNN marg.	0.393	0.426	0.994	0.552	0.694	0.732	1.169	0.783

Appendix

Table D.9: Exhaustive semisynthetic CATE nPEHE results at

n_{\mathrm{ctx}}=1,024

. Each entry first averages the three replicates of each source dataset, then averages equally over the 12 sources; lower is better. The lowest mean in each column is bold. Other underlined methods are not significantly worse than that empirical best under one-sided exact paired sign-flip tests across source-level means, with Holm correction within the column (

\alpha=0.05

).

	Linear				Nonlinear
	Low confounding		High confounding		Low confounding		High confounding
Method	Low proxy	High proxy	Low proxy	High proxy	Low proxy	High proxy	Low proxy	High proxy
S-learner (ridge)	1.011	1.011	1.560	1.560	1.022	1.022	1.606	1.606
S-learner (RF)	0.513	0.513	1.356	1.356	0.740	0.740	1.476	1.476
S-learner (TabICL)	0.310	0.310	1.239	1.239	0.564	0.564	1.371	1.371
S-learner (TabICL) + kNN marg.	0.291	0.298	0.953	0.441	0.580	0.624	1.111	0.695

Appendix

Table D.10: Exhaustive semisynthetic CATE nPEHE results at

n_{\mathrm{ctx}}=2,048

. Each entry first averages the three replicates of each source dataset, then averages equally over the 12 sources; lower is better. The lowest mean in each column is bold. Other underlined methods are not significantly worse than that empirical best under one-sided exact paired sign-flip tests across source-level means, with Holm correction within the column (

\alpha=0.05

).

Method	Context size $n_{\mathrm{ctx}}$
	$128$	$256$	$512$	$1,024$	$2,048$	$4,096$	$8,192$
S-learner (ridge)	1.459	1.499	1.580	1.593	1.606	1.597	1.608
S-learner (RF)	1.542	1.536	1.546	1.497	1.476	1.602	1.473
S-learner (TabICL)	1.457	1.449	1.432	1.393	1.371	1.340	1.317
S-learner (TabICL) + kNN marg.	1.259	1.235	1.206	1.169	1.111	1.065	1.020
S-learner (TabICL) + kernel marg.	1.245	1.231	1.196	1.159	1.111	1.064	1.016

Appendix

Table D.11: Extended-context semisynthetic CATE nPEHE results for nonlinear mechanisms with high confounding and low proxy reliability. Each entry first averages the 3 replicates of each source dataset, then averages equally over the 12 sources; lower is better. The lowest mean in each column is bold. Other underlined methods are not significantly worse than that empirical best under one-sided exact paired sign-flip tests across source-level means, with Holm correction within the column (

\alpha=0.05

).

Figure D.4: CATE RMSE against joint proxy information, measured by the population

R^{2}

for predicting

U

from

(W,Z)

. Columns vary proxy dimension

d

; rows vary context size. All methods are evaluated on the same 256 held-out query rows at each setting. The vertical axis is logarithmic. Within each dimension, the underlying draws are shared across information levels; results come from one realization per setting.

Figure E.1: The Light Tunnel Mk2 and a simplified diagram with the variables relevant to our setup.

Figure E.2: Physical and software pathways in the optical experiment. Solid arrows represent the predetermined physical pathways and dashed green arrows represent effects resulting from the software control functions, i.e., software mappings . Independent Gaussian noise enters the two proxy mappings. The lower sequence shows acquisition of the factual outcome and the two intervention measurements. The graph describes the intended within-observation structure. Its causal interpretation requires stable device state and the proxy restrictions discussed in the text.

Figure E.3: Software control functions for the physical experiment. Panels (a)–(c) show the deterministic mappings before addition of proxy noise and final actuator clipping. Panel (d) shows the treatment probabilities for Bernoulli assignment and the corresponding hard-threshold protocols. The plots display the specified control functions over illustrative sensor ranges.

Figure E.4: Absolute error of estimated average treatment effects in the Causal Chambers Light Tunnel experiment as the factual context grows. Each panel is one physical operating condition, crossing control-function family (linear or non-linear), confounding strength, and nominal proxy informativeness. Curves show

\lvert\widehat{\theta}_{n}-\widehat{\theta}_{\mathrm{ref},n}\rvert

in raw infrared-sensor counts, where both the estimate and the paired-intervention reference are evaluated on the same nested context prefix of size

n

. Both axes use logarithmic scales; lower is better.

ProximalFM: Amortized Proximal Causal Inference under Hidden Confounding

Abstract

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Appendix

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