We study partial identification of conditional treatment effects with a continuous treatment under unobserved confounding. We target the conditional average potential outcome (CAPO) E[Y_x|z] and the conditional average treatment effect (CATE) between two treatment levels. We assume a bounded outcome and Lipschitz continuity of the response functions and of the propensity scores given the latent confounder, and show that these conditions imply the smooth treatment response and smooth treatment selection conditions of Kim et al. (2018). The combined bound of Kim et al. uses only two reference points on the identified diagonal and is generally loose. Bounding through every diagonal reference point instead gives explicit bounds whose average is the sharp identified set of the CAPO at every treatment, inside or outside the observational support. Capping the difference between the upper and lower bounds gives sharp causal bounds on the CATE. These bounds are informative without monotonicity or overlap assumptions.
Figures & tables
Figure 1: Two confounded systems that induce the same observational law P(X,Y) ( blue ; regression E[Y∣x] in dark blue ) yet different interventional laws Px(Y) ( orange ), illustrating non-identifiability: (a) M with Y←−X2+U2 has a causal effect Ex[Y]=1−x2 ( dark orange curve); (b) M′ with Y←X2+21U2 has a causal effect Ex[Y]=x2+21 ( dark orange curve).
Figure 2: Half-width of the bound on the ETT over the (x,x′) plane for Exs. 2 and 3 . The identified diagonal x=x′ is dashed, and arrows point from each query to the diagonal point bounding it. (a) STR bound ( (STR) ); (b) STS bound ( (STS) ); (c) sharp bound over all diagonal reference points (Thm. 10 ); (d) enlarging a finite reference set I shrinks the averaged width toward the sharp bound. Panels (a)–(c) saturate at 21diam(Y)=10 .
Figure 3: The L-shaped path of Lem. 5 (blue) from (x,x′) through (s,x′) to the diagonal point (s,s) , at cost ρ(s,x′) (Eq. 14 ). A zig-zag path (grey) costs at least as much; query x may lie outside the support.
x′
s=0
s=1
s=2
two-point
sharp
0
2
1
2
2
1
1
3
0
1
0
0
2
4
1
0
0
0
Table 1: The counterexample of Prop. 9 at the query x=2 : terms μ(s)+∣2−s∣+∣x′−s∣ , the two-point bound of Kim et al. (2018) (columns s∈{2,x′} ), and the sharp bound. Averaging over P gives 2/3 and 1/3 .
Ours
Kim et al.
CMSM oracle
Khan et al.
Reduction
Setting
cov.
width
cov.
width
cov.
width
cov.
width
vs. Kim (%)
Discrete SCMs
1.00
0.11181
1.00
0.13832
1.00
0.04810
0.14
0.00000
19.17
Continuous SCMs
1.00
0.10627
1.00
0.13071
1.00
0.04841
0.04
0.00000
18.70
IST
1.00
0.03726
1.00
0.03777
1.00
0.07558
0.04
0.00000
1.34
Table 2: Query-averaged coverage and interval width normalized by the outcome range. Reduction compares pooled mean widths with Kim. CMSM uses a per-model coverage oracle, so its coverage holds by construction.
Figure 4: Example bounds and true means for discrete, smooth V-shaped continuous, and controlled IST models. Wine and gray shading denote ours and oracle CMSM; dashed and dotted boundaries denote Kim and CMSM. Discrete intervals use bars. Panels have separate outcome scales; example selection is specified in App. C .
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Setting
SCMs
Queries
Ours width
Kim width
Reduction (%)
Coverage (both)
Canonical
1
7
0.1119
0.1391
19.57
1.00/1.00
Random
500
3544
0.1118
0.1383
19.17
1.00/1.00
Appendix
Table 3: Stronger discrete variant with a V-shaped response. Widths are divided by the attained outcome range, diam(Y)=2 . Ours and Kim use identical exact structural constants; all random draws are retained.
Figure 5: Discrete V-shaped SCMs with matched structural constants and exact outcome-range clipping. Left: the canonical V-shaped response, its confounded observational regression, and both intervals. Middle: mean-width reductions for all 500 random systems. Right: the reduction across the assignment-tilt sweep, including settings where the methods tie.
Figure 6: Continuous V-shaped and smooth V-shaped response families. Top: canonical bounds at identical exact structural constants. Bottom: mean-width reductions against Kim across all 250 random draws per family, including every draw without rejection.
Age
Ours width
Kim width
Reduction (%)
Ours cov.
Kim cov.
16–67
0.01972
0.01981
0.47
1.00
1.00
68–77
0.04844
0.04909
1.32
1.00
1.00
78–99
0.04362
0.04440
1.76
1.00
1.00
Appendix
Table 4: Controlled IST-derived binary SCM: mean widths and coverage at identical oracle STR/STS constants. Each age group has 25 treatment queries. This model has a new, explicitly constructed target.
Figure 7: Controlled IST-derived binary SCM, with identical oracle STR/STS constants for both methods. Top: bounds, constructed interventional targets, and observational means in each age group. Bottom: the reduction in mean width as global references are added to the per-pair endpoints.
Setting
Family
Ours
Kim
Ours cov.
Kim cov.
Reduction (%)
Canonical
Polynomial
0.3480
0.3639
1.00
1.00
4.37
Canonical
Logistic
0.3243
0.3282
1.00
1.00
1.18
Canonical
Phase
0.2136
0.2279
1.00
1.00
6.29
Canonical
Curved linear
0.1316
0.1362
1.00
1.00
3.36
Random
Polynomial
0.3471
0.3616
1.00
1.00
4.01
Random
Logistic
0.3235
0.3271
1.00
1.00
1.08
Appendix
Table 5: Designed smooth continuous SCMs. Widths are normalized by each system’s outcome range. Random rows retain all 125 draws per family; reduction compares pooled mean widths. Canonical settings and neighborhoods were chosen after exploratory checks.
Figure 8: Designed smooth continuous SCMs with identical structural constants and outcome ranges for both methods. Each panel shows our full-reference bounds (dark red), Kim’s bounds (purple dashed), the true mean (orange), and the observational regression (blue). Top: polynomial and logistic; bottom: phase and curved linear. These are fixed canonical examples chosen after exploratory checks; Table 5 also reports all random draws in the stated neighborhoods.
References / method
Lower
Upper
Width
Coverage
Kim et al.
-1.000
0.667
1.667
1.00
Ours: endpoints
-1.000
0.667
1.667
1.00
Ours: endpoints + {0}
-1.000
0.667
1.667
1.00
Ours: endpoints + {0,1}
-1.000
0.333
1.333
1.00
Ours: all references
-1.000
0.333
1.333
1.00
Appendix
Table 6: Controlled reference ablation: fixed observational law, a=b=1 , and known outcome range [−2,2] . Endpoints are {x,x′} per stratum. Bounds and widths are at x=2 in outcome units; coverage is over all three queries. Adding s=1 reduces width by 20% without losing coverage.
Figure 9: Controlled reference ablation with identical data and constants. Left: Kim’s intervals and our full-reference intervals at all three queries, with the true means marked. Right: the interval at x=2 as the extra global reference set R grows. The interior reference s=1 lowers the upper endpoint to the attained sharp value 1/3 ; adding the remaining endpoint does not tighten it further.
We study the role of design smoothness in the minimax estimation of conditional average treatment effects at a fixed covariate point. For design densities bounded above and away from zero, without further regularity, we derive upper and lower risk bounds with an arbitrarily small gap in the convergence-rate exponent. Under uniform design, we obtain matching upper and lower bounds at interior points in the regime of low nuisance smoothness. These results show that rough covariate designs can lead to slower rates than uniform designs, even under the same smoothness assumptions on the treatment effect and nuisance functions, highlighting the role of the covariate distribution in pointwise causal estimation.
Yuxuan Hou, Zhenghan Li
Qiuzhen College, Tsinghua University · School of Statistics, University of Minnesota
Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in settings with discrete treatment and discrete ordinal outcome. We propose new families of monotonicity assumptions and formulate the bounding problem as a linear programming problem. We further introduce a new monotonicity assumption specifically to achieve identification. Finally, we present numerical experiments to validate our methods and demonstrate their application using real-world datasets.
Naoya Hashimoto, Yuta Kawakami, Jin Tian
Mohamed bin Zayed University of Artificial Intelligence
Unobserved confounding is a key challenge when estimating causal effects from a treatment on an outcome. In this work, we assume that we observe a single, potentially multi-dimensional proxy variable of the unobserved confounder and that we know the mechanism that generates the proxy from the confounder. Under an assumption called Single Proxy Identifiability of Causal Effects or simply SPICE, we prove that this error mechanism is complete and causal effects are identifiable. We extend the proxy-based causal identifiability results by Kuroki and Pearl (2014); Pearl (2010) to multi-dimensional continuous settings, more flexible functional relationships and a broader class of distributions. Further, we develop a neural network based estimation framework, SPICE-Net, to estimate causal effects, which is applicable to both discrete and continuous treatments.
Silvan Vollmer, Niklas Pfister, Sebastian Weichwald
Department of Mathematical Sciences University of Copenhagen Copenhagen, Denmark · Lakera AI Zurich, Switzerland