HiSTOR: Structured Treatment-outcome Representation Learning for High-Dimensional Longitudinal Data

Chuanqi Lao; Junzhou Huang; Qiao Liu

cs.LGOct 13, 2026

HiSTOR: Structured Treatment-outcome Representation Learning for High-Dimensional Longitudinal Data

Authors: Chuanqi Lao, Junzhou Huang, Qiao Liu

Organizations: Department of Computer Science and Engineering, The University of Texas at Arlington · Department of Biostatistics, Yale School of Public Health, Yale University

Abstract

Estimating potential outcomes under alternative treatment sequences is essential for personalized decision-making, but remains challenging with high-dimensional covariates and time-varying confounding. In this study, we propose HiSTOR (High-dimensional Structured Treatment-Outcome Representation), an end-to-end framework that integrates nonlinear dimension reduction with longitudinal counterfactual prediction. The central idea is to learn structured low-dimensional representations of both current patient states and evolving histories, according to their shared and distinct relationships with treatment assignment and outcomes. This extends latent-space covariate adjustment to longitudinal settings, jointly addressing covariate dimensionality and historical complexity. The resulting framework supports one-step and multi-step outcome prediction under specified treatment sequences. Experiments on CVSim and CancerSim demonstrate the superiority of HiSTOR across varying covariate dimensions, confounding strengths, and prediction horizons compared to state-of-the-art methods. To sum up, HiSTOR offers a unified approach to connecting high-dimensional covariate adjustment with longitudinal potential-outcome prediction.

Figures & tables

Figure 1: Overview of HiSTOR. The model adopts a temporal Transformer backbone and combines a high-dimensional current patient state with a learned representation of longitudinal history, maps their fusion into a structured latent space, and supports treatment-conditioned flexible one-step and multi-step outcome prediction.

Method	$\gamma$ = 0	$\gamma$ = 1	$\gamma$ = 2	$\gamma$ = 3	$\gamma$ = 4
MSM ( Robins et al., 2000 )	1.0298 $\pm$ 0.0633	1.1312 $\pm$ 0.0633	1.3005 $\pm$ 0.1039	1.6059 $\pm$ 0.1681	2.0113 $\pm$ 0.2446
RMSN ( Lim, 2018 )	0.6994 $\pm$ 0.0690	0.6789 $\pm$ 0.0758	0.7446 $\pm$ 0.1153	0.7324 $\pm$ 0.1042	0.8256 $\pm$ 0.0850
CRN ( Bica et al., 2020 )	0.5969 $\pm$ 0.0639	0.6098 $\pm$ 0.0480	0.6492 $\pm$ 0.0585	0.7143 $\pm$ 0.0868	0.8256 $\pm$ 0.0994
CT ( Melnychuk et al., 2022 )	0.5928 $\pm$ 0.0665	0.5974 $\pm$ 0.0855	0.6390 $\pm$ 0.0612	0.7372 $\pm$ 0.0954	0.9335 $\pm$ 0.0748
Causal CPC ( Bouchattaoui et al., 2024 )	0.6142 $\pm$ 0.0535	0.6228 $\pm$ 0.0636	0.6922 $\pm$ 0.1313	0.7235 $\pm$ 0.0767	0.8427 $\pm$ 0.0857
CTD-NKO ( Wang et al., 2025 )	0.6852 $\pm$ 0.0825	0.6454 $\pm$ 0.1011	0.6659 $\pm$ 0.1056	0.7122 $\pm$ 0.0390	0.7620 $\pm$ 0.0566

Table 1: The results for CancerSim one-step prediction. Mean

\pm

sample SD of NRMSE over five seeds are shown. Best means are bold, second-best underlined; the final row reports HiSTOR’s relative reduction over the best baseline.

Method	$\gamma$ = 0	$\gamma$ = 1	$\gamma$ = 2	$\gamma$ = 3	$\gamma$ = 4
MSM ( Robins et al., 2000 )	1.0170 $\pm$ 0.0600	0.8856 $\pm$ 0.0410	0.8822 $\pm$ 0.0224	0.9833 $\pm$ 0.0533	1.2124 $\pm$ 0.0645
RMSN ( Lim, 2018 )	0.5735 $\pm$ 0.0637	0.5968 $\pm$ 0.0515	0.6584 $\pm$ 0.0373	0.8303 $\pm$ 0.0851	1.0953 $\pm$ 0.0821
CRN ( Bica et al., 2020 )	0.5573 $\pm$ 0.0157	0.5951 $\pm$ 0.0317	0.6946 $\pm$ 0.0353	0.8111 $\pm$ 0.0463	1.0908 $\pm$ 0.0937
CT ( Melnychuk et al., 2022 )	0.5381 $\pm$ 0.0399	0.5776 $\pm$ 0.0315	0.6619 $\pm$ 0.0549	0.8657 $\pm$ 0.0510	1.1267 $\pm$ 0.1961
Causal CPC ( Bouchattaoui et al., 2024 )	0.5374 $\pm$ 0.0366	0.5865 $\pm$ 0.0547	0.6494 $\pm$ 0.0218	0.7934 $\pm$ 0.0543	1.0701 $\pm$ 0.0575
CTD-NKO ( Wang et al., 2025 )	0.5462 $\pm$ 0.0502	0.5664 $\pm$ 0.0295	0.6473 $\pm$ 0.0514	0.7997 $\pm$ 0.0993	1.2394 $\pm$ 0.1614

Table 2: The results for CancerSim multi-step prediction averaged over

\tau=2

6

. Mean

\pm

sample SD of NRMSE over five seeds are shown. Best means are bold, second-best underlined; the final row reports HiSTOR’s relative reduction over the best baseline.

Figure 2: CancerSim sustained-treatment NRMSE over

\tau=2,\ldots,6

. Lines are five-seed means; the gray envelope is the lowest mean among all baselines at each horizon and may switch methods. Complete results and SDs are in Appendix G .

Figure 3: CVSim NRMSE across nine

d_{\mathrm{hd}}

settings. Panels (a,b) show Static and panels (c,d) Temporal results; the left panels report independent one-step error and the right panels report five-horizon means. Lines are five-seed means. Appendix Figure 4 includes all methods.

CVSim Static
Method	One-step	Multi-step
Method	One-step	$\tau=1$	$\tau=2$	$\tau=3$	$\tau=4$	$\tau=5$
MSM ( Robins et al., 2000 )	2.797 $\pm$ 1.659	3.797 $\pm$ 2.357	3.860 $\pm$ 2.392	3.914 $\pm$ 2.419	4.017 $\pm$ 2.479	4.082 $\pm$ 2.515
RMSN ( Lim, 2018 )	2.857 $\pm$ 0.975	3.775 $\pm$ 1.181	3.202 $\pm$ 0.735	3.147 $\pm$ 0.639	3.323 $\pm$ 0.591	3.544 $\pm$ 0.553
CRN ( Bica et al., 2020 )	4.359 $\pm$ 0.909	6.076 $\pm$ 1.675	7.004 $\pm$ 1.878	7.669 $\pm$ 1.911	8.433 $\pm$ 1.907	9.192 $\pm$ 1.945
CT ( Melnychuk et al., 2022 )	1.175 $\pm$ 0.318	1.412 $\pm$ 0.454	1.846 $\pm$ 0.472	2.234 $\pm$ 0.802	2.632 $\pm$ 1.150	2.984 $\pm$ 1.342

Table 3: CVSim NRMSE at

d_{\mathrm{hd}}=2000

, reported as mean

\pm

sample SD over five seeds. Independent one-step and multi-step

\tau=1

are distinct tasks. Best means are bold, second-best underlined; relative improvements are computed against the lowest-mean baseline in each column.

Variant	$\gamma=1$		$\gamma=4$
Variant	One-step	Five-horizon mean	One-step	Five-horizon mean
HiSTOR	$\mathbf{0.5663\pm 0.0607}$	$\mathbf{0.5291\pm 0.0173}$	$\mathbf{0.6703\pm 0.0451}$	$\mathbf{0.9105\pm 0.0208}$
w/o Covariate Bottleneck	$0.5928\pm 0.0500$	$0.5664\pm 0.0495$	$0.6797\pm 0.0356$	$0.9555\pm 0.0349$
w/o Distribution Matching	$0.5777\pm 0.0622$	$0.5516\pm 0.0205$	$0.6805\pm 0.0336$	$0.9189\pm 0.0498$
w/o Reconstruction	$0.5771\pm 0.0524$	$0.5492\pm 0.0282$	$0.6813\pm 0.0554$	$0.9151\pm 0.0570$
Autoregressive Prediction	—	$0.5502\pm 0.0210$	—	$0.9193\pm 0.0409$

Table 4: CancerSim ablation NRMSE, reported as mean

\pm

sample SD over five seeds. Bold values indicate the lowest mean in each column. One-step results for autoregressive prediction are omitted because it uses the same frozen backbone as HiSTOR.

Appendix figures & tables42 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
$X_{t}$	Treatment assigned at time $t$
$Y_{t}$	Post-treatment outcome associated with $X_{t}$
$V_{t}$	Covariates available before treatment at time $t$ , including baseline and, when present, time-varying components
$\mathcal{H}_{t}$	Observed information available before treatment at time $t$
$W_{t}$	Implementation-only notation for the dataset-specific raw input supplied to the current-state projection $P$
$S$	Static CancerSim patient-type feature; the baseline component of $V_{t}$ used through the static-feature channel

Appendix

Table 5: Core notation.

Parameter	Value
Patients; train/validation/test	5,000; 4,000/500/500
Trajectory length; state dimension	50; 6
Base time-varying/static covariates	4/3
Treatment categories	0, 1, 2, encoded as three-way one-hot vectors
Integration step; confounding coefficient	0.1; 1.0
State/covariate/outcome noise SD	0.05/0.05/0.05

Appendix

Table 6: CVSim simulation and evaluation parameters used in the experiments.

Parameter	Value
Training/validation patients	10,000/1,000
Patients per counterfactual test cohort	1,000 for each separately generated test
Maximum simulation length	60; at most 59 shifted factual prediction slots
Treatment categories	None, chemotherapy, radiotherapy, both
Static covariate	Patient type, sampled uniformly from $\{1,2,3\}$
Chemo/radio assignment coefficients	Both equal $\gamma\in\{0,1,2,3,4\}$

Appendix

Table 7: CancerSim simulation and evaluation parameters used in the experiments. Sampling means and standard deviations refer to the parameter distributions before truncation.

Method	Multi-step inputs
HiSTOR	Direct decoding uses the boundary representation, static covariates, previous-outcome input, and treatment plan, without future covariates or outcome feedback between horizon heads. CancerSim first supplies a predicted bridge outcome at the boundary.
CT ( Melnychuk et al., 2022 )	Masks access to post-origin covariates and recursively updates outcomes.
CRN, RMSN	Decode from the encoded history using treatment, static information, and predicted outcomes; decoders do not consume future covariates.
MSM ( Robins et al., 2000 )	Uses origin-history features and cumulative treatment-plan features, without future-covariate simulation.
Causal CPC ( Bouchattaoui et al., 2024 )	Decodes from origin context and the treatment plan, without future covariates.
CTD-NKO ( Wang et al., 2025 )	Recursively predicts outcomes and subsequent covariates, replacing placeholders.

Appendix

Table 8: Future-covariate handling within the common forecast boundary.

Method	Maximum budget	Validation and checkpoint selection
CRN ( Bica et al., 2020 )	100 epochs per encoder/decoder stage	Fixed budget; no best-validation restoration
CT ( Melnychuk et al., 2022 )	150 epochs	Fixed budget; final EMA parameters
RMSN ( Lim, 2018 )	100 epochs per propensity/prediction stage	Fixed budget; no best-validation restoration
HiSTOR (backbone)	CancerSim: 50 epochs; CVSim: 120 epochs	EMA validation. CancerSim: one-step MSE; CVSim: one-step MSE plus $0.25$ times rollout MSE. Patience: 12/25, respectively.
HiSTOR (decoder)	200 epochs	Factual multi-step loss in Equation 25 ; no EMA; patience 20.
MSM ( Robins et al., 2000 )	Weighted regression	Training-set fit; no validation checkpoint

Appendix

Table 9: Training budgets and validation criteria.

Method	$\gamma$ =0	$\gamma$ =1	$\gamma$ =2	$\gamma$ =3	$\gamma$ =4
MSM ( Robins et al., 2000 )	1.2741 $\pm$ 0.1100	1.1635 $\pm$ 0.0900	1.2066 $\pm$ 0.0585	1.4065 $\pm$ 0.1020	1.8154 $\pm$ 0.1062
RMSN ( Lim, 2018 )	0.9211 $\pm$ 0.1186	0.9329 $\pm$ 0.0732	1.0331 $\pm$ 0.0750	1.1923 $\pm$ 0.1337	1.5835 $\pm$ 0.1700
CRN ( Bica et al., 2020 )	0.9276 $\pm$ 0.0552	0.9578 $\pm$ 0.0771	1.0822 $\pm$ 0.0609	1.1791 $\pm$ 0.0744	1.4956 $\pm$ 0.1417
CT ( Melnychuk et al., 2022 )	0.8526 $\pm$ 0.0666	0.8956 $\pm$ 0.0510	1.0073 $\pm$ 0.0781	1.2217 $\pm$ 0.1078	1.5822 $\pm$ 0.1954
Causal CPC ( Bouchattaoui et al., 2024 )	0.8707 $\pm$ 0.0642	0.9144 $\pm$ 0.0661	1.0265 $\pm$ 0.0297	1.1786 $\pm$ 0.0823	1.5764 $\pm$ 0.0844
CTD-NKO ( Wang et al., 2025 )	0.8988 $\pm$ 0.0903	0.8836 $\pm$ 0.0466	0.9827 $\pm$ 0.0878	1.1608 $\pm$ 0.1459	1.5919 $\pm$ 0.1812

Appendix

Table 10: CancerSim:

\tau

=2. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	$\gamma$ =0	$\gamma$ =1	$\gamma$ =2	$\gamma$ =3	$\gamma$ =4
MSM ( Robins et al., 2000 )	1.2151 $\pm$ 0.0802	1.0676 $\pm$ 0.0612	1.0760 $\pm$ 0.0378	1.2109 $\pm$ 0.0858	1.4938 $\pm$ 0.0915
RMSN ( Lim, 2018 )	0.7531 $\pm$ 0.0863	0.7798 $\pm$ 0.0677	0.8688 $\pm$ 0.0547	1.0709 $\pm$ 0.0985	1.4446 $\pm$ 0.1342
CRN ( Bica et al., 2020 )	0.7207 $\pm$ 0.0200	0.7605 $\pm$ 0.0470	0.8918 $\pm$ 0.0493	1.0124 $\pm$ 0.0468	1.3358 $\pm$ 0.1304
CT ( Melnychuk et al., 2022 )	0.7039 $\pm$ 0.0512	0.7565 $\pm$ 0.0371	0.8613 $\pm$ 0.0584	1.1286 $\pm$ 0.0820	1.4427 $\pm$ 0.1822
Causal CPC ( Bouchattaoui et al., 2024 )	0.6892 $\pm$ 0.0442	0.7593 $\pm$ 0.0648	0.8426 $\pm$ 0.0366	1.0072 $\pm$ 0.0809	1.3924 $\pm$ 0.0735
CTD-NKO ( Wang et al., 2025 )	0.7168 $\pm$ 0.0679	0.7402 $\pm$ 0.0398	0.8424 $\pm$ 0.0726	1.0433 $\pm$ 0.1289	1.5613 $\pm$ 0.1790

Appendix

Table 11: CancerSim:

\tau

=3. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	$\gamma$ =0	$\gamma$ =1	$\gamma$ =2	$\gamma$ =3	$\gamma$ =4
MSM ( Robins et al., 2000 )	1.0396 $\pm$ 0.0558	0.8940 $\pm$ 0.0395	0.8815 $\pm$ 0.0251	0.9675 $\pm$ 0.0506	1.1679 $\pm$ 0.0738
RMSN ( Lim, 2018 )	0.5467 $\pm$ 0.0640	0.5764 $\pm$ 0.0653	0.6356 $\pm$ 0.0360	0.8279 $\pm$ 0.0876	1.1093 $\pm$ 0.0867
CRN ( Bica et al., 2020 )	0.5204 $\pm$ 0.0256	0.5637 $\pm$ 0.0280	0.6723 $\pm$ 0.0483	0.7932 $\pm$ 0.0311	1.0614 $\pm$ 0.0893
CT ( Melnychuk et al., 2022 )	0.5144 $\pm$ 0.0404	0.5589 $\pm$ 0.0314	0.6427 $\pm$ 0.0474	0.8771 $\pm$ 0.0610	1.1302 $\pm$ 0.2003
Causal CPC ( Bouchattaoui et al., 2024 )	0.5041 $\pm$ 0.0368	0.5608 $\pm$ 0.0545	0.6149 $\pm$ 0.0279	0.7665 $\pm$ 0.0606	1.0591 $\pm$ 0.0822
CTD-NKO ( Wang et al., 2025 )	0.5135 $\pm$ 0.0504	0.5486 $\pm$ 0.0352	0.6347 $\pm$ 0.0528	0.7999 $\pm$ 0.1004	1.2832 $\pm$ 0.1673

Appendix

Table 12: CancerSim:

\tau

=4. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	$\gamma$ =0	$\gamma$ =1	$\gamma$ =2	$\gamma$ =3	$\gamma$ =4
MSM ( Robins et al., 2000 )	0.8566 $\pm$ 0.0448	0.7227 $\pm$ 0.0299	0.6987 $\pm$ 0.0291	0.7520 $\pm$ 0.0293	0.8948 $\pm$ 0.0575
RMSN ( Lim, 2018 )	0.3820 $\pm$ 0.0486	0.4087 $\pm$ 0.0528	0.4453 $\pm$ 0.0248	0.6126 $\pm$ 0.0932	0.7896 $\pm$ 0.0523
CRN ( Bica et al., 2020 )	0.3643 $\pm$ 0.0262	0.4064 $\pm$ 0.0229	0.4838 $\pm$ 0.0341	0.6091 $\pm$ 0.0470	0.8571 $\pm$ 0.0685
CT ( Melnychuk et al., 2022 )	0.3630 $\pm$ 0.0327	0.3966 $\pm$ 0.0269	0.4629 $\pm$ 0.0467	0.6433 $\pm$ 0.0502	0.8458 $\pm$ 0.2130
Causal CPC ( Bouchattaoui et al., 2024 )	0.3610 $\pm$ 0.0287	0.4055 $\pm$ 0.0513	0.4418 $\pm$ 0.0222	0.5710 $\pm$ 0.0431	0.7627 $\pm$ 0.0573
CTD-NKO ( Wang et al., 2025 )	0.3553 $\pm$ 0.0348	0.3872 $\pm$ 0.0264	0.4547 $\pm$ 0.0322	0.5801 $\pm$ 0.0754	0.9987 $\pm$ 0.1647

Appendix

Table 13: CancerSim:

\tau

=5. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	$\gamma$ =0	$\gamma$ =1	$\gamma$ =2	$\gamma$ =3	$\gamma$ =4
MSM ( Robins et al., 2000 )	0.6995 $\pm$ 0.0387	0.5805 $\pm$ 0.0272	0.5482 $\pm$ 0.0285	0.5792 $\pm$ 0.0194	0.6899 $\pm$ 0.0397
RMSN ( Lim, 2018 )	0.2648 $\pm$ 0.0380	0.2863 $\pm$ 0.0380	0.3091 $\pm$ 0.0188	0.4479 $\pm$ 0.1021	0.5498 $\pm$ 0.0400
CRN ( Bica et al., 2020 )	0.2533 $\pm$ 0.0236	0.2874 $\pm$ 0.0165	0.3428 $\pm$ 0.0225	0.4616 $\pm$ 0.0873	0.7041 $\pm$ 0.0592
CT ( Melnychuk et al., 2022 )	0.2566 $\pm$ 0.0286	0.2806 $\pm$ 0.0225	0.3352 $\pm$ 0.0513	0.4578 $\pm$ 0.0380	0.6323 $\pm$ 0.2171
Causal CPC ( Bouchattaoui et al., 2024 )	0.2622 $\pm$ 0.0182	0.2925 $\pm$ 0.0479	0.3213 $\pm$ 0.0170	0.4434 $\pm$ 0.0416	0.5598 $\pm$ 0.0440
CTD-NKO ( Wang et al., 2025 )	0.2466 $\pm$ 0.0237	0.2723 $\pm$ 0.0190	0.3219 $\pm$ 0.0182	0.4144 $\pm$ 0.0566	0.7617 $\pm$ 0.1655

Appendix

Table 14: CancerSim:

\tau

=6. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Condition	Comparator	Wins / 5	Mean difference	Approx. 95% interval
0	CT ( Melnychuk et al., 2022 )	3/5	-0.0248	[-0.0693, 0.0198]
1	CT ( Melnychuk et al., 2022 )	4/5	-0.0311	[-0.0811, 0.0188]
2	CT ( Melnychuk et al., 2022 )	5/5	-0.0597	[-0.1243, 0.0049]
3	CTD-NKO ( Wang et al., 2025 )	5/5	-0.0957	[-0.1435, -0.0478]
4	CTD-NKO ( Wang et al., 2025 )	5/5	-0.0917	[-0.1622, -0.0212]

Appendix

Table 15: CancerSim: paired comparisons for independent one-step.

Condition	Comparator	Wins / 5	Mean difference	Approx. 95% interval
0	Causal CPC ( Bouchattaoui et al., 2024 )	5/5	-0.0351	[-0.0633, -0.0068]
1	CTD-NKO ( Wang et al., 2025 )	5/5	-0.0373	[-0.0650, -0.0096]
2	CTD-NKO ( Wang et al., 2025 )	5/5	-0.0509	[-0.1024, 0.0007]
3	Causal CPC ( Bouchattaoui et al., 2024 )	5/5	-0.0873	[-0.1167, -0.0579]
4	Causal CPC ( Bouchattaoui et al., 2024 )	5/5	-0.1596	[-0.2107, -0.1085]

Appendix

Table 16: CancerSim: paired comparisons for five-horizon averages.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	2.7450 $\pm$ 1.6322	2.7454 $\pm$ 1.6333	2.7450 $\pm$ 1.6337	2.7442 $\pm$ 1.6323	2.7449 $\pm$ 1.6336
RMSN ( Lim, 2018 )	2.6288 $\pm$ 1.4077	2.5587 $\pm$ 1.3994	2.5495 $\pm$ 1.3605	2.5482 $\pm$ 1.3358	2.5416 $\pm$ 1.3063
CRN ( Bica et al., 2020 )	4.3197 $\pm$ 3.1278	3.4662 $\pm$ 1.5178	3.4285 $\pm$ 1.5090	3.0455 $\pm$ 1.3404	2.9070 $\pm$ 1.3528
CT ( Melnychuk et al., 2022 )	1.1909 $\pm$ 0.2306	1.1130 $\pm$ 0.2309	1.0561 $\pm$ 0.2623	1.0544 $\pm$ 0.2537	1.1011 $\pm$ 0.2754
Causal CPC ( Bouchattaoui et al., 2024 )	1.0229 $\pm$ 0.2806	0.9878 $\pm$ 0.1462	0.9647 $\pm$ 0.2157	1.0222 $\pm$ 0.1746	1.0315 $\pm$ 0.1334
CTD-NKO ( Wang et al., 2025 )	2.6155 $\pm$ 1.3320	2.6028 $\pm$ 1.2879	2.6830 $\pm$ 1.3248	2.7466 $\pm$ 1.3316	2.6789 $\pm$ 1.3221

Appendix

Table 17: CVSim Static: independent one-step. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	2.7480 $\pm$ 1.6364	2.7470 $\pm$ 1.6352	2.7533 $\pm$ 1.6372	2.7967 $\pm$ 1.6590
RMSN ( Lim, 2018 )	2.5427 $\pm$ 1.2593	2.4430 $\pm$ 1.1556	2.4987 $\pm$ 1.0909	2.8571 $\pm$ 0.9745
CRN ( Bica et al., 2020 )	2.9344 $\pm$ 1.2817	3.4432 $\pm$ 1.2395	5.0170 $\pm$ 2.5278	4.3593 $\pm$ 0.9090
CT ( Melnychuk et al., 2022 )	1.1546 $\pm$ 0.2004	1.0759 $\pm$ 0.2817	1.1336 $\pm$ 0.3511	1.1746 $\pm$ 0.3178
Causal CPC ( Bouchattaoui et al., 2024 )	0.9441 $\pm$ 0.1871	1.0135 $\pm$ 0.1543	1.1242 $\pm$ 0.2058	1.2622 $\pm$ 0.3074
CTD-NKO ( Wang et al., 2025 )	2.8848 $\pm$ 1.5157	3.1853 $\pm$ 1.6830	3.1071 $\pm$ 1.5156	3.3701 $\pm$ 1.7194

Appendix

Table 18: CVSim Static: independent one-step. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	3.8995 $\pm$ 2.3936	3.9023 $\pm$ 2.3966	3.9011 $\pm$ 2.4019	3.8970 $\pm$ 2.3968	3.9030 $\pm$ 2.4009
RMSN ( Lim, 2018 )	1.4049 $\pm$ 0.2490	1.3870 $\pm$ 0.2611	1.3362 $\pm$ 0.3567	1.5987 $\pm$ 0.4111	1.5739 $\pm$ 0.2969
CRN ( Bica et al., 2020 )	9.0673 $\pm$ 7.7702	8.9171 $\pm$ 7.0460	8.8059 $\pm$ 6.8482	8.5169 $\pm$ 6.0701	6.9780 $\pm$ 4.4332
CT ( Melnychuk et al., 2022 )	2.0750 $\pm$ 0.4091	1.9703 $\pm$ 0.5882	1.8430 $\pm$ 0.4586	1.8120 $\pm$ 0.4352	2.1216 $\pm$ 0.6573
Causal CPC ( Bouchattaoui et al., 2024 )	1.2583 $\pm$ 0.2243	1.3926 $\pm$ 0.2146	1.3327 $\pm$ 0.3708	1.4799 $\pm$ 0.2834	1.4441 $\pm$ 0.1880
CTD-NKO ( Wang et al., 2025 )	1.3025 $\pm$ 0.3854	1.2620 $\pm$ 0.4424	1.1858 $\pm$ 0.5011	1.5834 $\pm$ 0.6597	1.5202 $\pm$ 0.6123

Appendix

Table 19: CVSim Static: H=5 mean. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	3.9029 $\pm$ 2.3989	3.8979 $\pm$ 2.4004	3.8804 $\pm$ 2.3867	3.9338 $\pm$ 2.4313
RMSN ( Lim, 2018 )	1.6446 $\pm$ 0.3517	1.8023 $\pm$ 0.4819	2.2660 $\pm$ 0.6327	3.3983 $\pm$ 0.6877
CRN ( Bica et al., 2020 )	7.4061 $\pm$ 4.4954	6.6797 $\pm$ 3.4750	10.1151 $\pm$ 5.9721	7.6749 $\pm$ 1.7319
CT ( Melnychuk et al., 2022 )	2.2070 $\pm$ 0.5525	2.3559 $\pm$ 0.6702	2.0990 $\pm$ 0.4857	2.2217 $\pm$ 0.7037
Causal CPC ( Bouchattaoui et al., 2024 )	1.3728 $\pm$ 0.3327	1.3718 $\pm$ 0.3357	1.5421 $\pm$ 0.6982	1.6714 $\pm$ 0.3533
CTD-NKO ( Wang et al., 2025 )	2.2378 $\pm$ 1.1573	3.9033 $\pm$ 2.6208	3.0568 $\pm$ 1.5703	4.4738 $\pm$ 2.0124

Appendix

Table 20: CVSim Static: H=5 mean. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	3.7967 $\pm$ 2.3417	3.7980 $\pm$ 2.3431	3.7982 $\pm$ 2.3480	3.7950 $\pm$ 2.3471	3.7989 $\pm$ 2.3487
RMSN ( Lim, 2018 )	1.1531 $\pm$ 0.2963	1.0590 $\pm$ 0.2679	1.0448 $\pm$ 0.1897	1.2750 $\pm$ 0.4344	1.3775 $\pm$ 0.4149
CRN ( Bica et al., 2020 )	6.8521 $\pm$ 6.2222	4.8815 $\pm$ 3.3568	4.8326 $\pm$ 3.4881	4.0465 $\pm$ 3.2664	3.8699 $\pm$ 3.3554
CT ( Melnychuk et al., 2022 )	1.4014 $\pm$ 0.3895	1.3001 $\pm$ 0.4927	1.2009 $\pm$ 0.3307	1.1995 $\pm$ 0.3709	1.2679 $\pm$ 0.3902
Causal CPC ( Bouchattaoui et al., 2024 )	1.1443 $\pm$ 0.2723	1.1734 $\pm$ 0.3190	1.2380 $\pm$ 0.6390	1.2519 $\pm$ 0.3046	1.2238 $\pm$ 0.4068
CTD-NKO ( Wang et al., 2025 )	1.0182 $\pm$ 0.3781	0.9375 $\pm$ 0.1359	0.8845 $\pm$ 0.2905	1.2035 $\pm$ 0.1932	1.3646 $\pm$ 0.2754

Appendix

Table 21: CVSim Static:

\tau

=1. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	3.8005 $\pm$ 2.3501	3.7934 $\pm$ 2.3489	3.7636 $\pm$ 2.3257	3.7965 $\pm$ 2.3568
RMSN ( Lim, 2018 )	1.4634 $\pm$ 0.3948	1.6731 $\pm$ 0.2485	2.3968 $\pm$ 0.7055	3.7753 $\pm$ 1.1812
CRN ( Bica et al., 2020 )	4.0863 $\pm$ 3.4004	4.5051 $\pm$ 3.0349	8.0181 $\pm$ 5.7618	6.0755 $\pm$ 1.6745
CT ( Melnychuk et al., 2022 )	1.4265 $\pm$ 0.3914	1.4002 $\pm$ 0.5101	1.3635 $\pm$ 0.4614	1.4122 $\pm$ 0.4542
Causal CPC ( Bouchattaoui et al., 2024 )	1.2106 $\pm$ 0.4059	1.2280 $\pm$ 0.4627	1.4765 $\pm$ 0.9703	1.5326 $\pm$ 0.6243
CTD-NKO ( Wang et al., 2025 )	1.8459 $\pm$ 1.1898	2.8919 $\pm$ 1.5128	2.6533 $\pm$ 1.1288	3.8557 $\pm$ 1.4492

Appendix

Table 22: CVSim Static:

\tau

=1. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	3.8421 $\pm$ 2.3669	3.8444 $\pm$ 2.3688	3.8435 $\pm$ 2.3744	3.8398 $\pm$ 2.3717	3.8436 $\pm$ 2.3733
RMSN ( Lim, 2018 )	1.2818 $\pm$ 0.3601	1.3209 $\pm$ 0.3758	1.2316 $\pm$ 0.4053	1.5392 $\pm$ 0.5251	1.4706 $\pm$ 0.3845
CRN ( Bica et al., 2020 )	8.7891 $\pm$ 7.7591	9.0417 $\pm$ 8.4202	9.2318 $\pm$ 8.6043	8.8772 $\pm$ 7.9215	6.3405 $\pm$ 5.1047
CT ( Melnychuk et al., 2022 )	1.7815 $\pm$ 0.5207	1.6361 $\pm$ 0.5089	1.5811 $\pm$ 0.2875	1.5177 $\pm$ 0.3577	1.7857 $\pm$ 0.4054
Causal CPC ( Bouchattaoui et al., 2024 )	1.2125 $\pm$ 0.1918	1.3220 $\pm$ 0.2847	1.2781 $\pm$ 0.5236	1.4102 $\pm$ 0.3295	1.3547 $\pm$ 0.2720
CTD-NKO ( Wang et al., 2025 )	1.1625 $\pm$ 0.3745	1.1364 $\pm$ 0.2813	1.0632 $\pm$ 0.3760	1.2666 $\pm$ 0.4762	1.2273 $\pm$ 0.5151

Appendix

Table 23: CVSim Static:

\tau

=2. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	3.8473 $\pm$ 2.3735	3.8390 $\pm$ 2.3698	3.8171 $\pm$ 2.3576	3.8602 $\pm$ 2.3920
RMSN ( Lim, 2018 )	1.5285 $\pm$ 0.4599	1.6396 $\pm$ 0.6311	2.0595 $\pm$ 0.7100	3.2020 $\pm$ 0.7347
CRN ( Bica et al., 2020 )	7.0946 $\pm$ 5.1845	5.9010 $\pm$ 3.9557	9.5498 $\pm$ 6.1773	7.0041 $\pm$ 1.8783
CT ( Melnychuk et al., 2022 )	1.8717 $\pm$ 0.3589	1.9612 $\pm$ 0.6664	1.7431 $\pm$ 0.5346	1.8464 $\pm$ 0.4721
Causal CPC ( Bouchattaoui et al., 2024 )	1.2941 $\pm$ 0.3640	1.3024 $\pm$ 0.4044	1.5344 $\pm$ 0.8348	1.5859 $\pm$ 0.5107
CTD-NKO ( Wang et al., 2025 )	1.7462 $\pm$ 1.0320	3.0734 $\pm$ 2.2712	2.5635 $\pm$ 1.4733	3.8654 $\pm$ 1.9996

Appendix

Table 24: CVSim Static:

\tau

=2. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	3.8858 $\pm$ 2.3833	3.8889 $\pm$ 2.3869	3.8878 $\pm$ 2.3928	3.8827 $\pm$ 2.3867	3.8893 $\pm$ 2.3900
RMSN ( Lim, 2018 )	1.3476 $\pm$ 0.3118	1.3437 $\pm$ 0.2847	1.2530 $\pm$ 0.4230	1.5526 $\pm$ 0.4808	1.5011 $\pm$ 0.3281
CRN ( Bica et al., 2020 )	9.1894 $\pm$ 8.0217	9.4800 $\pm$ 8.2640	9.4051 $\pm$ 8.1390	9.3331 $\pm$ 7.6721	7.1943 $\pm$ 4.7858
CT ( Melnychuk et al., 2022 )	2.0565 $\pm$ 0.4461	1.9468 $\pm$ 0.6155	1.8455 $\pm$ 0.4525	1.7779 $\pm$ 0.4644	2.1193 $\pm$ 0.6504
Causal CPC ( Bouchattaoui et al., 2024 )	1.2392 $\pm$ 0.1970	1.3901 $\pm$ 0.2311	1.3172 $\pm$ 0.3776	1.4882 $\pm$ 0.3049	1.4384 $\pm$ 0.1717
CTD-NKO ( Wang et al., 2025 )	1.2602 $\pm$ 0.3715	1.2056 $\pm$ 0.4227	1.1264 $\pm$ 0.4706	1.4315 $\pm$ 0.6483	1.3177 $\pm$ 0.6717

Appendix

Table 25: CVSim Static:

\tau

=3. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	3.8885 $\pm$ 2.3890	3.8831 $\pm$ 2.3888	3.8620 $\pm$ 2.3730	3.9135 $\pm$ 2.4187
RMSN ( Lim, 2018 )	1.5693 $\pm$ 0.4167	1.6966 $\pm$ 0.5911	2.0625 $\pm$ 0.6452	3.1471 $\pm$ 0.6391
CRN ( Bica et al., 2020 )	7.7512 $\pm$ 5.0555	6.7121 $\pm$ 3.7592	10.1678 $\pm$ 6.1497	7.6694 $\pm$ 1.9110
CT ( Melnychuk et al., 2022 )	2.2215 $\pm$ 0.5228	2.4012 $\pm$ 0.7244	2.0839 $\pm$ 0.4917	2.2336 $\pm$ 0.8017
Causal CPC ( Bouchattaoui et al., 2024 )	1.3440 $\pm$ 0.3049	1.3528 $\pm$ 0.3272	1.5384 $\pm$ 0.6873	1.6396 $\pm$ 0.3721
CTD-NKO ( Wang et al., 2025 )	2.0075 $\pm$ 1.1139	3.6483 $\pm$ 2.8250	2.8035 $\pm$ 1.7030	4.2220 $\pm$ 2.2681

Appendix

Table 26: CVSim Static:

\tau

=3. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	3.9651 $\pm$ 2.4204	3.9685 $\pm$ 2.4244	3.9672 $\pm$ 2.4295	3.9625 $\pm$ 2.4224	3.9699 $\pm$ 2.4303
RMSN ( Lim, 2018 )	1.5065 $\pm$ 0.3061	1.4922 $\pm$ 0.2924	1.4398 $\pm$ 0.4848	1.7078 $\pm$ 0.5304	1.6583 $\pm$ 0.3616
CRN ( Bica et al., 2020 )	9.8884 $\pm$ 8.3587	10.2238 $\pm$ 8.2640	9.9621 $\pm$ 7.8215	9.8489 $\pm$ 7.2351	8.2260 $\pm$ 4.7884
CT ( Melnychuk et al., 2022 )	2.4058 $\pm$ 0.4334	2.3078 $\pm$ 0.8270	2.1282 $\pm$ 0.7222	2.0975 $\pm$ 0.6891	2.5231 $\pm$ 1.0198
Causal CPC ( Bouchattaoui et al., 2024 )	1.2969 $\pm$ 0.3309	1.4873 $\pm$ 0.2119	1.3800 $\pm$ 0.2703	1.5829 $\pm$ 0.3423	1.5449 $\pm$ 0.1957
CTD-NKO ( Wang et al., 2025 )	1.4428 $\pm$ 0.4619	1.4238 $\pm$ 0.6436	1.3305 $\pm$ 0.6556	1.8057 $\pm$ 0.9354	1.6440 $\pm$ 0.7936

Appendix

Table 27: CVSim Static:

\tau

=4. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	3.9677 $\pm$ 2.4229	3.9630 $\pm$ 2.4252	3.9557 $\pm$ 2.4191	4.0165 $\pm$ 2.4791
RMSN ( Lim, 2018 )	1.7159 $\pm$ 0.4682	1.8696 $\pm$ 0.5355	2.2626 $\pm$ 0.6166	3.3231 $\pm$ 0.5906
CRN ( Bica et al., 2020 )	8.6021 $\pm$ 5.1523	7.6696 $\pm$ 3.5421	11.0051 $\pm$ 6.0536	8.4334 $\pm$ 1.9068
CT ( Melnychuk et al., 2022 )	2.5971 $\pm$ 0.8808	2.8316 $\pm$ 0.8829	2.4722 $\pm$ 0.6265	2.6322 $\pm$ 1.1500
Causal CPC ( Bouchattaoui et al., 2024 )	1.4506 $\pm$ 0.3698	1.4478 $\pm$ 0.3257	1.5702 $\pm$ 0.5720	1.7409 $\pm$ 0.2619
CTD-NKO ( Wang et al., 2025 )	2.5020 $\pm$ 1.2278	4.4983 $\pm$ 3.1942	3.3130 $\pm$ 1.8087	4.8590 $\pm$ 2.2819

Appendix

Table 28: CVSim Static:

\tau

=4. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	4.0077 $\pm$ 2.4594	4.0114 $\pm$ 2.4640	4.0090 $\pm$ 2.4684	4.0053 $\pm$ 2.4597	4.0131 $\pm$ 2.4662
RMSN ( Lim, 2018 )	1.7355 $\pm$ 0.2844	1.7191 $\pm$ 0.3702	1.7118 $\pm$ 0.5419	1.9189 $\pm$ 0.6002	1.8623 $\pm$ 0.4085
CRN ( Bica et al., 2020 )	10.6173 $\pm$ 8.6792	10.9587 $\pm$ 8.2702	10.5977 $\pm$ 7.6693	10.4788 $\pm$ 7.0183	9.2596 $\pm$ 4.9251
CT ( Melnychuk et al., 2022 )	2.7297 $\pm$ 0.5142	2.6609 $\pm$ 1.0704	2.4593 $\pm$ 1.0107	2.4672 $\pm$ 0.9946	2.9122 $\pm$ 1.3927
Causal CPC ( Bouchattaoui et al., 2024 )	1.3985 $\pm$ 0.4466	1.5903 $\pm$ 0.3243	1.4500 $\pm$ 0.2503	1.6662 $\pm$ 0.4207	1.6587 $\pm$ 0.3049
CTD-NKO ( Wang et al., 2025 )	1.6286 $\pm$ 0.5091	1.6069 $\pm$ 0.8057	1.5242 $\pm$ 0.7642	2.2097 $\pm$ 1.1367	2.0473 $\pm$ 0.8866

Appendix

Table 29: CVSim Static:

\tau

=5. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	4.0103 $\pm$ 2.4627	4.0108 $\pm$ 2.4733	4.0036 $\pm$ 2.4620	4.0823 $\pm$ 2.5153
RMSN ( Lim, 2018 )	1.9458 $\pm$ 0.5239	2.1325 $\pm$ 0.5349	2.5487 $\pm$ 0.5972	3.5440 $\pm$ 0.5534
CRN ( Bica et al., 2020 )	9.4963 $\pm$ 5.3358	8.6107 $\pm$ 3.3439	11.8347 $\pm$ 5.9853	9.1921 $\pm$ 1.9449
CT ( Melnychuk et al., 2022 )	2.9184 $\pm$ 1.2140	3.1855 $\pm$ 1.1446	2.8321 $\pm$ 0.8347	2.9842 $\pm$ 1.3423
Causal CPC ( Bouchattaoui et al., 2024 )	1.5645 $\pm$ 0.4738	1.5278 $\pm$ 0.3469	1.5910 $\pm$ 0.4851	1.8580 $\pm$ 0.3081
CTD-NKO ( Wang et al., 2025 )	3.0872 $\pm$ 1.3923	5.4045 $\pm$ 3.4026	3.9509 $\pm$ 1.8503	5.5671 $\pm$ 2.1347

Appendix

Table 30: CVSim Static:

\tau

=5. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	2.7450 $\pm$ 1.6322	2.7454 $\pm$ 1.6333	2.7452 $\pm$ 1.6325	2.7459 $\pm$ 1.6326	2.7468 $\pm$ 1.6339
RMSN ( Lim, 2018 )	2.6294 $\pm$ 1.4075	2.5317 $\pm$ 1.3379	2.4788 $\pm$ 1.3331	2.4691 $\pm$ 1.3105	2.4451 $\pm$ 1.1880
CRN ( Bica et al., 2020 )	4.4422 $\pm$ 3.0405	3.0922 $\pm$ 0.9791	2.8961 $\pm$ 1.2966	2.8790 $\pm$ 1.6229	2.9901 $\pm$ 1.1956
CT ( Melnychuk et al., 2022 )	1.2015 $\pm$ 0.2318	1.1074 $\pm$ 0.2483	1.0451 $\pm$ 0.2390	1.0620 $\pm$ 0.2633	1.0627 $\pm$ 0.2857
Causal CPC ( Bouchattaoui et al., 2024 )	1.0343 $\pm$ 0.2127	1.0016 $\pm$ 0.1462	0.9737 $\pm$ 0.2364	0.9262 $\pm$ 0.1202	0.9020 $\pm$ 0.2280
CTD-NKO ( Wang et al., 2025 )	2.6337 $\pm$ 1.3748	2.6658 $\pm$ 1.4017	2.6573 $\pm$ 1.3121	2.7368 $\pm$ 1.3708	2.7254 $\pm$ 1.4472

Appendix

Table 31: CVSim Temporal: independent one-step. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	2.7473 $\pm$ 1.6328	2.7503 $\pm$ 1.6370	2.7513 $\pm$ 1.6350	2.7764 $\pm$ 1.6611
RMSN ( Lim, 2018 )	2.4139 $\pm$ 1.1749	2.4714 $\pm$ 1.1313	2.4826 $\pm$ 1.0474	2.6053 $\pm$ 0.9674
CRN ( Bica et al., 2020 )	2.7721 $\pm$ 1.1417	4.2994 $\pm$ 3.0813	3.8956 $\pm$ 2.0132	4.6170 $\pm$ 2.4998
CT ( Melnychuk et al., 2022 )	1.2468 $\pm$ 0.2041	1.0792 $\pm$ 0.3036	1.1018 $\pm$ 0.3079	1.1316 $\pm$ 0.2828
Causal CPC ( Bouchattaoui et al., 2024 )	0.8554 $\pm$ 0.2100	1.0186 $\pm$ 0.1984	1.0259 $\pm$ 0.2398	1.0417 $\pm$ 0.1852
CTD-NKO ( Wang et al., 2025 )	2.6362 $\pm$ 1.3536	3.3635 $\pm$ 1.7934	3.3747 $\pm$ 1.7954	3.3819 $\pm$ 1.7992

Appendix

Table 32: CVSim Temporal: independent one-step. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	3.8995 $\pm$ 2.3936	3.8986 $\pm$ 2.3975	3.8989 $\pm$ 2.3969	3.9016 $\pm$ 2.3974	3.9049 $\pm$ 2.3950
RMSN ( Lim, 2018 )	1.4081 $\pm$ 0.2477	1.4633 $\pm$ 0.3766	1.3780 $\pm$ 0.3220	1.5407 $\pm$ 0.3899	1.6851 $\pm$ 0.4357
CRN ( Bica et al., 2020 )	9.1398 $\pm$ 7.6974	8.5451 $\pm$ 6.8844	8.6751 $\pm$ 6.9349	8.8647 $\pm$ 7.0063	7.2686 $\pm$ 4.7127
CT ( Melnychuk et al., 2022 )	2.1296 $\pm$ 0.5086	2.0102 $\pm$ 0.6012	1.7952 $\pm$ 0.3714	1.8452 $\pm$ 0.4301	1.9914 $\pm$ 0.4479
Causal CPC ( Bouchattaoui et al., 2024 )	1.3727 $\pm$ 0.3482	1.8688 $\pm$ 0.9381	1.6531 $\pm$ 0.5848	1.3039 $\pm$ 0.3076	1.2967 $\pm$ 0.2788
CTD-NKO ( Wang et al., 2025 )	1.3442 $\pm$ 0.2660	1.2410 $\pm$ 0.4004	1.2593 $\pm$ 0.5008	1.3771 $\pm$ 0.4523	1.4741 $\pm$ 0.5179

Appendix

Table 33: CVSim Temporal: H=5 mean. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	3.9050 $\pm$ 2.4022	3.9088 $\pm$ 2.4049	3.9184 $\pm$ 2.3946	3.9638 $\pm$ 2.4453
RMSN ( Lim, 2018 )	1.6362 $\pm$ 0.4110	1.7675 $\pm$ 0.4534	2.0422 $\pm$ 0.5464	2.6230 $\pm$ 0.8039
CRN ( Bica et al., 2020 )	8.5728 $\pm$ 7.5055	9.2303 $\pm$ 8.2751	8.4620 $\pm$ 6.3207	9.1284 $\pm$ 6.2666
CT ( Melnychuk et al., 2022 )	2.1523 $\pm$ 0.7620	2.0136 $\pm$ 0.5755	2.0252 $\pm$ 0.4911	1.9134 $\pm$ 0.3355
Causal CPC ( Bouchattaoui et al., 2024 )	1.1721 $\pm$ 0.2252	1.7428 $\pm$ 0.4955	1.4770 $\pm$ 0.8099	1.3624 $\pm$ 0.4074
CTD-NKO ( Wang et al., 2025 )	1.6892 $\pm$ 0.6190	4.7669 $\pm$ 2.4956	4.8129 $\pm$ 2.5124	4.8762 $\pm$ 2.5523

Appendix

Table 34: CVSim Temporal: H=5 mean. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	3.7967 $\pm$ 2.3417	3.7962 $\pm$ 2.3460	3.7948 $\pm$ 2.3411	3.7981 $\pm$ 2.3436	3.8022 $\pm$ 2.3439
RMSN ( Lim, 2018 )	1.1525 $\pm$ 0.2959	1.0561 $\pm$ 0.2378	1.0494 $\pm$ 0.1727	1.0915 $\pm$ 0.1531	1.4266 $\pm$ 0.3521
CRN ( Bica et al., 2020 )	7.1539 $\pm$ 6.0200	4.2965 $\pm$ 2.6300	4.0356 $\pm$ 3.2368	4.0400 $\pm$ 3.4593	3.9995 $\pm$ 3.0871
CT ( Melnychuk et al., 2022 )	1.4235 $\pm$ 0.4274	1.3013 $\pm$ 0.4951	1.2217 $\pm$ 0.3652	1.2406 $\pm$ 0.3828	1.2940 $\pm$ 0.4059
Causal CPC ( Bouchattaoui et al., 2024 )	1.2283 $\pm$ 0.3749	1.3479 $\pm$ 0.5813	1.4328 $\pm$ 0.8236	1.0898 $\pm$ 0.4290	1.1082 $\pm$ 0.3818
CTD-NKO ( Wang et al., 2025 )	1.0944 $\pm$ 0.3759	0.8241 $\pm$ 0.2657	0.8813 $\pm$ 0.2566	1.0029 $\pm$ 0.2229	1.1181 $\pm$ 0.2819

Appendix

Table 35: CVSim Temporal:

\tau

=1. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	3.8028 $\pm$ 2.3540	3.8076 $\pm$ 2.3731	3.8032 $\pm$ 2.3484	3.8414 $\pm$ 2.4254
RMSN ( Lim, 2018 )	1.4443 $\pm$ 0.1762	1.9093 $\pm$ 0.4264	2.3739 $\pm$ 0.7301	3.0402 $\pm$ 1.3213
CRN ( Bica et al., 2020 )	3.7383 $\pm$ 2.5857	6.9846 $\pm$ 7.1065	5.6170 $\pm$ 3.8932	7.5510 $\pm$ 5.7171
CT ( Melnychuk et al., 2022 )	1.4020 $\pm$ 0.3368	1.2786 $\pm$ 0.4567	1.3430 $\pm$ 0.4789	1.3580 $\pm$ 0.4511
Causal CPC ( Bouchattaoui et al., 2024 )	1.0631 $\pm$ 0.3207	1.4190 $\pm$ 0.7191	1.4839 $\pm$ 1.2221	1.2567 $\pm$ 0.5121
CTD-NKO ( Wang et al., 2025 )	1.4214 $\pm$ 0.3317	4.4267 $\pm$ 2.1838	4.4615 $\pm$ 2.2147	4.4542 $\pm$ 2.2475

Appendix

Table 36: CVSim Temporal:

\tau

=1. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	3.8421 $\pm$ 2.3669	3.8411 $\pm$ 2.3713	3.8418 $\pm$ 2.3708	3.8452 $\pm$ 2.3705	3.8503 $\pm$ 2.3724
RMSN ( Lim, 2018 )	1.2856 $\pm$ 0.3597	1.4236 $\pm$ 0.4497	1.3069 $\pm$ 0.3973	1.4910 $\pm$ 0.4588	1.5823 $\pm$ 0.4680
CRN ( Bica et al., 2020 )	8.7599 $\pm$ 7.7889	9.1807 $\pm$ 8.6508	9.1705 $\pm$ 8.6337	9.0367 $\pm$ 8.3286	6.8978 $\pm$ 5.0452
CT ( Melnychuk et al., 2022 )	1.8273 $\pm$ 0.5952	1.6233 $\pm$ 0.4889	1.5592 $\pm$ 0.3426	1.5558 $\pm$ 0.4011	1.6731 $\pm$ 0.4366
Causal CPC ( Bouchattaoui et al., 2024 )	1.3387 $\pm$ 0.3295	1.5632 $\pm$ 0.6230	1.5250 $\pm$ 0.6959	1.1963 $\pm$ 0.3568	1.1807 $\pm$ 0.3038
CTD-NKO ( Wang et al., 2025 )	1.2266 $\pm$ 0.2930	1.0614 $\pm$ 0.3058	1.0858 $\pm$ 0.3369	1.1267 $\pm$ 0.2747	1.2568 $\pm$ 0.3553

Appendix

Table 37: CVSim Temporal:

\tau

=2. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	3.8427 $\pm$ 2.3760	3.8495 $\pm$ 2.3678	3.8605 $\pm$ 2.3732	3.9045 $\pm$ 2.4321
RMSN ( Lim, 2018 )	1.4734 $\pm$ 0.4912	1.5514 $\pm$ 0.5455	1.8450 $\pm$ 0.5639	2.4118 $\pm$ 0.7545
CRN ( Bica et al., 2020 )	8.4191 $\pm$ 8.0628	8.7964 $\pm$ 8.6260	8.1064 $\pm$ 7.0506	8.6584 $\pm$ 6.4498
CT ( Melnychuk et al., 2022 )	1.8379 $\pm$ 0.3928	1.6826 $\pm$ 0.5708	1.7027 $\pm$ 0.5107	1.5954 $\pm$ 0.3979
Causal CPC ( Bouchattaoui et al., 2024 )	1.1090 $\pm$ 0.2561	1.5502 $\pm$ 0.6138	1.4870 $\pm$ 1.0508	1.2529 $\pm$ 0.4568
CTD-NKO ( Wang et al., 2025 )	1.4866 $\pm$ 0.4620	4.4460 $\pm$ 2.3377	4.4858 $\pm$ 2.3760	4.4887 $\pm$ 2.3739

Appendix

Table 38: CVSim Temporal:

\tau

=2. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	3.8858 $\pm$ 2.3833	3.8842 $\pm$ 2.3868	3.8865 $\pm$ 2.3897	3.8888 $\pm$ 2.3902	3.8908 $\pm$ 2.3856
RMSN ( Lim, 2018 )	1.3522 $\pm$ 0.3108	1.4307 $\pm$ 0.4132	1.3081 $\pm$ 0.3365	1.5213 $\pm$ 0.4338	1.6321 $\pm$ 0.4570
CRN ( Bica et al., 2020 )	9.2023 $\pm$ 7.9992	9.1453 $\pm$ 8.1649	9.5191 $\pm$ 8.6350	9.8500 $\pm$ 8.8859	7.6501 $\pm$ 5.3291
CT ( Melnychuk et al., 2022 )	2.1180 $\pm$ 0.5476	1.9765 $\pm$ 0.6233	1.7822 $\pm$ 0.3857	1.8253 $\pm$ 0.5196	1.9645 $\pm$ 0.4525
Causal CPC ( Bouchattaoui et al., 2024 )	1.3751 $\pm$ 0.3206	1.8174 $\pm$ 0.8584	1.6337 $\pm$ 0.5918	1.2919 $\pm$ 0.2888	1.2509 $\pm$ 0.2657
CTD-NKO ( Wang et al., 2025 )	1.3109 $\pm$ 0.2354	1.1867 $\pm$ 0.3959	1.1934 $\pm$ 0.4943	1.2935 $\pm$ 0.3924	1.3953 $\pm$ 0.4537

Appendix

Table 39: CVSim Temporal:

\tau

=3. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	3.8903 $\pm$ 2.3920	3.8871 $\pm$ 2.3959	3.9055 $\pm$ 2.3782	3.9438 $\pm$ 2.4199
RMSN ( Lim, 2018 )	1.4991 $\pm$ 0.4947	1.5643 $\pm$ 0.4692	1.7588 $\pm$ 0.5120	2.3022 $\pm$ 0.7176
CRN ( Bica et al., 2020 )	9.6523 $\pm$ 9.7523	9.3451 $\pm$ 8.5753	8.7543 $\pm$ 6.9779	9.1869 $\pm$ 6.4209
CT ( Melnychuk et al., 2022 )	2.1612 $\pm$ 0.7570	2.0144 $\pm$ 0.5815	2.0017 $\pm$ 0.5027	1.8937 $\pm$ 0.3048
Causal CPC ( Bouchattaoui et al., 2024 )	1.1478 $\pm$ 0.2038	1.7079 $\pm$ 0.4869	1.4826 $\pm$ 0.8399	1.3182 $\pm$ 0.4269
CTD-NKO ( Wang et al., 2025 )	1.6351 $\pm$ 0.6281	4.6420 $\pm$ 2.5112	4.6873 $\pm$ 2.5473	4.7475 $\pm$ 2.5618

Appendix

Table 40: CVSim Temporal:

\tau

=3. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	3.9651 $\pm$ 2.4204	3.9649 $\pm$ 2.4240	3.9650 $\pm$ 2.4230	3.9674 $\pm$ 2.4246	3.9681 $\pm$ 2.4180
RMSN ( Lim, 2018 )	1.5109 $\pm$ 0.3055	1.5847 $\pm$ 0.4538	1.4825 $\pm$ 0.4160	1.6825 $\pm$ 0.4671	1.7795 $\pm$ 0.5406
CRN ( Bica et al., 2020 )	9.9220 $\pm$ 8.2963	9.7230 $\pm$ 8.2491	10.0333 $\pm$ 8.3287	10.4043 $\pm$ 8.5707	8.4971 $\pm$ 5.6567
CT ( Melnychuk et al., 2022 )	2.4662 $\pm$ 0.5405	2.3808 $\pm$ 0.8863	2.0439 $\pm$ 0.5551	2.1270 $\pm$ 0.6799	2.3368 $\pm$ 0.5892
Causal CPC ( Bouchattaoui et al., 2024 )	1.4283 $\pm$ 0.3899	2.1444 $\pm$ 1.2535	1.7726 $\pm$ 0.5054	1.4170 $\pm$ 0.2990	1.3949 $\pm$ 0.3476
CTD-NKO ( Wang et al., 2025 )	1.4702 $\pm$ 0.3164	1.4211 $\pm$ 0.5684	1.4438 $\pm$ 0.7029	1.5732 $\pm$ 0.6627	1.6515 $\pm$ 0.7137

Appendix

Table 41: CVSim Temporal:

\tau

=4. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	3.9756 $\pm$ 2.4257	3.9841 $\pm$ 2.4437	3.9866 $\pm$ 2.4153	4.0472 $\pm$ 2.4655
RMSN ( Lim, 2018 )	1.7232 $\pm$ 0.5150	1.7660 $\pm$ 0.4591	1.9739 $\pm$ 0.5195	2.5343 $\pm$ 0.6988
CRN ( Bica et al., 2020 )	10.2041 $\pm$ 9.6734	10.1112 $\pm$ 8.5561	9.5223 $\pm$ 6.9038	9.7942 $\pm$ 6.4114
CT ( Melnychuk et al., 2022 )	2.5202 $\pm$ 1.1858	2.3838 $\pm$ 0.7752	2.3744 $\pm$ 0.6647	2.2099 $\pm$ 0.6148
Causal CPC ( Bouchattaoui et al., 2024 )	1.2229 $\pm$ 0.2336	1.9155 $\pm$ 0.4871	1.4785 $\pm$ 0.6079	1.4279 $\pm$ 0.4289
CTD-NKO ( Wang et al., 2025 )	1.8271 $\pm$ 0.8175	4.9769 $\pm$ 2.6887	5.0271 $\pm$ 2.7036	5.1400 $\pm$ 2.7537

Appendix

Table 42: CVSim Temporal:

\tau

=4. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=0	d_hd=10	d_hd=20	d_hd=50	d_hd=100
MSM ( Robins et al., 2000 )	4.0077 $\pm$ 2.4594	4.0064 $\pm$ 2.4631	4.0066 $\pm$ 2.4641	4.0085 $\pm$ 2.4621	4.0131 $\pm$ 2.4587
RMSN ( Lim, 2018 )	1.7393 $\pm$ 0.2836	1.8216 $\pm$ 0.5468	1.7433 $\pm$ 0.4956	1.9173 $\pm$ 0.5048	2.0049 $\pm$ 0.6214
CRN ( Bica et al., 2020 )	10.6612 $\pm$ 8.5815	10.3802 $\pm$ 8.4132	10.6169 $\pm$ 8.1565	10.9928 $\pm$ 8.2859	9.2986 $\pm$ 5.9158
CT ( Melnychuk et al., 2022 )	2.8132 $\pm$ 0.6403	2.7692 $\pm$ 1.1529	2.3687 $\pm$ 0.8105	2.4770 $\pm$ 0.9247	2.6887 $\pm$ 0.8144
Causal CPC ( Bouchattaoui et al., 2024 )	1.4933 $\pm$ 0.4758	2.4711 $\pm$ 1.7521	1.9015 $\pm$ 0.4874	1.5247 $\pm$ 0.3522	1.5489 $\pm$ 0.4284
CTD-NKO ( Wang et al., 2025 )	1.6191 $\pm$ 0.4061	1.7119 $\pm$ 0.5881	1.6924 $\pm$ 0.8664	1.8892 $\pm$ 0.9017	1.9486 $\pm$ 0.9485

Appendix

Table 43: CVSim Temporal:

\tau

=5. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Method	d_hd=200	d_hd=500	d_hd=1000	d_hd=2000
MSM ( Robins et al., 2000 )	4.0137 $\pm$ 2.4674	4.0157 $\pm$ 2.4475	4.0362 $\pm$ 2.4616	4.0822 $\pm$ 2.4862
RMSN ( Lim, 2018 )	2.0413 $\pm$ 0.5540	2.0463 $\pm$ 0.4792	2.2591 $\pm$ 0.5369	2.8261 $\pm$ 0.6786
CRN ( Bica et al., 2020 )	10.8501 $\pm$ 9.6670	10.9141 $\pm$ 8.5901	10.3101 $\pm$ 6.9036	10.4517 $\pm$ 6.4611
CT ( Melnychuk et al., 2022 )	2.8403 $\pm$ 1.5287	2.7089 $\pm$ 1.1283	2.7040 $\pm$ 0.8821	2.5102 $\pm$ 0.9041
Causal CPC ( Bouchattaoui et al., 2024 )	1.3179 $\pm$ 0.2902	2.1217 $\pm$ 0.6315	1.4528 $\pm$ 0.3846	1.5564 $\pm$ 0.4550
CTD-NKO ( Wang et al., 2025 )	2.0755 $\pm$ 0.9644	5.3428 $\pm$ 2.7960	5.4029 $\pm$ 2.7734	5.5504 $\pm$ 2.8749

Appendix

Table 44: CVSim Temporal:

\tau

=5. Values are NRMSE, mean

\pm

sample SD across five randomly selected seeds

\{3,101,42,8,9\}

. Lower is better; the lowest mean is bold and the second-lowest underlined.

Figure 4: All eight methods across the complete CVSim dimension grid. Lines show five-seed mean NRMSE.

Method	$\tau=1$	$\tau=2$	$\tau=3$	$\tau=4$	$\tau=5$	$\tau=6$
CT ( Melnychuk et al., 2022 )	0.1269	0.3157	0.4134	0.4717	0.5256	0.5824
IGC-Net ( Hess et al., 2026 )	0.1349	0.3537	0.4430	0.5299	0.5599	0.6519
HiSTOR	0.1701	0.3148	0.3951	0.4709	0.4949	0.5446

Appendix

Table 45: Values are mean scaled RMSE over seeds

\{3,101,42,8,9\}

; lower is better. Standard deviations are omitted from this compact table.

Causal Effect Estimation

Causal effect estimation measures how outcomes would change under interventions, using experiments or observational data and explicit assumptions.

See all 266 papers

Jun 4, 2026

Causal Longitudinal Prior-Fitted Networks for Counterfactual Outcome Prediction

Longitudinal treatment decisions from multivariate time-series data require predicting potential outcomes under future treatment sequences in the presence of time-varying confounding, heterogeneous patient dynamics, and limited domain-specific data. Existing longitudinal causal estimators typically address this problem by training a new model for each cohort or simulator. We introduce Causal Longitudinal Prior-Fitted Networks (CausalLongPFN), a prior-fitted network for time-series causal inference in longitudinal treatment-response data and zero-shot in-context counterfactual outcome prediction. The model is pretrained entirely on synthetic episodes sampled from a broad prior over temporal structural causal models, exposing it to treatment-confounder feedback, latent heterogeneity, nonlinear state evolution, delayed effects, and cumulative treatment responses. At test time, CausalLongPFN remains frozen and is used zero-shot: it conditions on support trajectories, a query history, and a planned future treatment sequence, and returns a predictive distribution over future outcomes without gradient updates or propensity-model fitting. Multi-step predictions are obtained by recursively applying the one-step predictor under the specified treatment sequence. We evaluate the model on branchable cancer, HIV, and warfarin benchmarks with ground-truth counterfactual labels, and on factual-only rolling-origin prediction in MIMIC-III ICU trajectories. CausalLongPFN is competitive with domain-trained longitudinal baselines on counterfactual benchmarks and performs strongly on factual MIMIC-III prediction, suggesting that broad synthetic causal pretraining can provide a frozen, amortized alternative for zero-shot longitudinal treatment-response prediction when repeated domain-specific training is costly or impractical.

Amirhossein Zare, Amirhessam Zare, Herlock Rahimi +2

School of Medicine, Tehran University of Medical Sciences, Tehran, Iran. · Department of Electrical and Computer Engineering, Yale University, New Haven, CT, USA. · School of Medicine, Shiraz University of Medical Sciences, Shiraz, Iran. +2

Structural Causal Models Counterfactual Prediction Prior-Data Fitted Networks Causal Inference Longitudinal EHR Modeling Counterfactual Learning

Ask PDF

Aug 8, 2026

Causal State-Space Model for Causal Inference: Estimating Longitudinal Individual Treatment Effects

Estimating counterfactual outcomes over time from longitudinal observational data is central to clinical decision support. Existing methods rely on domain confusion -- adversarial training that renders representations invariant to treatment assignment -- yet this invariance creates a mutual information conflict: it suppresses treatment-correlated covariate signals necessary for accurate outcome prediction. We formalise this tension via a Jensen-Shannon divergence bound on counterfactual prediction error and develop two complementary models. CSSD (Causal State-Space model with Direct decoder) adapts selective State Space Models with a parallel multi-step decoder that eliminates accumulated rollout error by producing all prediction horizons simultaneously in a single forward pass. CSSPD (Causal State-Space model with Predictive regularisation and Direct decoder) augments CSSD with Contrastive Predictive Coding and Local Information Maximisation to reinforce temporal predictability in the balancing representation and recover local covariate information destroyed by domain confusion. On MIMIC-III, CSSPD achieves lower counterfactual RMSE than the Causal Transformer at every horizon tau >= 2 at O(T) encoder cost, with gains from 0.02 (2-step) to 0.07 (6-step). On Cancer Simulation across confounding strengths gamma in {0,1,2,3,4}, CSSPD outperforms CT at gamma <= 3 (margins 25.9%--37.0%), and CSSD achieves the lowest overall average RMSE (12.7% reduction over CT), confirming the MI conflict analysis. To our knowledge, this is the first work to formalise the balancing-prediction MI conflict and propose a structured resolution through complementary predictive and information-theoretic training objectives.

Abisoye Abidakun, Mingjun Zhong, Georgios Leontidis

Department of Computing Science, University of Aberdeen, Aberdeen, UK · UiT The Arctic University of Norway

Causal Effect Estimation Selective SSMs Causal Counterfactual Generation Clinical Outcome Prediction Counterfactual Prediction Causal Inference+1

Ask PDF

Apr 14, 2026

Causal Diffusion Models for Counterfactual Outcome Distributions in Longitudinal Data

Predicting counterfactual outcomes in longitudinal data, where sequential treatment decisions heavily depend on evolving patient states, is critical yet notoriously challenging due to complex time-dependent confounding and inadequate uncertainty quantification in existing methods. We introduce the Causal Diffusion Model (CDM), the first denoising diffusion probabilistic approach explicitly designed to generate full probabilistic distributions of counterfactual outcomes under sequential interventions. CDM employs a novel residual denoising architecture with relational self-attention, capturing intricate temporal dependencies and multimodal outcome trajectories without requiring explicit adjustments (e.g., inverse-probability weighting or adversarial balancing) for confounding. In rigorous evaluation on a pharmacokinetic-pharmacodynamic tumor-growth simulator widely adopted in prior work, CDM consistently outperforms state-of-the-art longitudinal causal inference methods, achieving a 15-30% relative improvement in distributional accuracy (1-Wasserstein distance) while maintaining competitive or superior point-estimate accuracy (RMSE) under high-confounding regimes. By unifying uncertainty quantification and robust counterfactual prediction in complex, sequentially confounded settings, without tailored deconfounding, CDM offers a flexible, high-impact tool for decision support in medicine, policy evaluation, and other longitudinal domains.

Farbod Alinezhad, Jianfei Cao, Gary J. Young +1

Analysis Group · Department of Economics, Northeastern University · Center for Health Policy and Healthcare Research, Northeastern University +1

Counterfactual Prediction Causal Inference Diffusion Models Conditional Diffusion Models

Ask PDF

Abstract

Causal Effect Estimation

Explore similar work

Causal Longitudinal Prior-Fitted Networks for Counterfactual Outcome Prediction

Causal State-Space Model for Causal Inference: Estimating Longitudinal Individual Treatment Effects

Causal Diffusion Models for Counterfactual Outcome Distributions in Longitudinal Data