Bayesian Causal Forests for Zero-Inflated Continuous Outcomes

Hugo Gobato Souto; Eric Wang; Igor Shuryak

stat.MLOct 8, 2026

Bayesian Causal Forests for Zero-Inflated Continuous Outcomes

Authors: Hugo Gobato Souto, Eric Wang, Igor Shuryak

Organizations: University of São Paulo, Brazil. · Columbia University Irving Medical Center.

Abstract

Estimating (heterogeneous) treatment effects is still challenging when the outcome is semicontinuous (i.e., exhibiting both a discrete point mass at zero and a skewed continuous distribution for positive values), despite the strong presence of this data type in real world settings. Existing methods either ignore the zero mass, impose fragile parametric assumptions, or suffer from extreme computational costs and data augmentation noise. This paper proposes Zero-Inflated Continuous Bayesian Causal Forests with Duan's Smearing (ZIC-BCF-Smear), a decoupled two-part nonparametric estimator that fits a Probit BCF on the full sample to model participation and a Gaussian BCF strictly on the active subpopulation to model positive intensity. Subpopulation Propensity Adjustment (SPA) mitigates regularization-induced confounding within the active subpopulation, and Duan's non-parametric smearing estimator re-transforms predictions to the response scale, bypassing the fragile scale multipliers of parametric models. A comprehensive simulation benchmark shows that ZIC-BCF-Smear matches well-specified parametric models without their assumptions and remains stable and calibrated where competitors either diverge or pathologically over-cover for nonlinear data. The method is then applied to individual-participant data from the SPECTRUM randomized phase III head-and-neck cancer trial, estimating the causal effect of assignment to panitumumab plus chemotherapy on the number of days patients spend with severe toxicity, an outcome for which a quarter of patients contribute an exact zero. Panitumumab assignment raises that burden by about ten days per patient under the trial's own differential treatment duration, and the decoupled structure localizes the effect mainly in the positive-intensity part rather than in whether the reconstructed severe-AE burden is positive.

Figures & tables

Feature	BCF-Linear	DPglm	EDP+ZI	Gamma Hurdle	Gamma + .01	ZIC-BCF-Smear
Separates Zero-Inflation	No	Yes	Yes	Yes	No	Yes
Hurdle CATE Estimation	No	Yes	Yes	Yes	No	Yes
MCMC Augmentation Noise	None	Extreme	Extreme	None	None	None
Subpopulation Confounding	N/A	Adjusted	Adjusted	Adjusted	N/A	Adjusted (SPA)
Scale Re-transformation	None	Parametric	Parametric	Parametric	Parametric	Non-Parametric

Table 1: Qualitative comparison of semicontinuous causal estimators.

DGP	Model	CATE RMSE	CATE Cov	CATE Corr	CATE CI Len	ATE RMSE	ATE Abs Bias
DGP A: Log-Normal	BCF-Linear	3.662	93.2%	0.184	6.315	0.849	0.282
	DPglm	1.409	100.0%	0.321	51.210	0.781	0.637
	Gamma Hurdle	1.810	94.9%	0.419	4.641	0.585	0.029
	Gamma +.01	2.946	91.7%	0.267	6.215	0.765	0.212
	ZIC-BCF-Smear	1.526	94.9%	0.397	3.819	0.614	0.406
DGP B: Gamma	BCF-Linear	4.010	92.5%	0.156	6.489	0.812	0.253

Table 2: Performance metrics compiled across 100 independent seeds at the standard configuration (

N=500

, ZI Level 3,

\approx 40\%

zeros) under link-linear DGPs.

DGP	Model	CATE RMSE	CATE Cov	CATE Corr	CATE CI Len	ATE RMSE	ATE Abs Bias
DGP A: Log-Normal	BCF-Linear	5.812	88.1%	0.347	6.385	1.127	0.172
	DPglm	3.070	99.7%	0.198	61.532	0.680	0.021
	Gamma Hurdle	$9.7\times 10^{7}\dagger$	69.2%	-0.039	$5.8\times 10^{7}\dagger$	$5.6\times 10^{8}\dagger$	$5.9\times 10^{7}\dagger$
	Gamma +.01	$3.0\times 10^{9}\dagger$	69.5%	-0.081	$2.1\times 10^{11}\dagger$	$7.5\times 10^{8}\dagger$	$7.3\times 10^{7}\dagger$
	ZIC-BCF-Smear	1.734	89.7%	0.686	4.422	0.537	0.119
DGP B: Gamma	BCF-Linear	5.792	89.1%	0.318	6.624	1.101	0.217

Table 3: Standard comparative results (

N=500

, ZI Level 3,

\approx 40\%

zeros) under link-nonlinear DGPs. Dagger (

\dagger

) indicates metrics contaminated by numerical divergence.

		Link-Linear		Link-Nonlinear
DGP	Model	Hurdle RMSE	Coverage	Hurdle RMSE	Coverage
DGP A: Log-Normal	DPglm	0.053	99.7%	0.077	98.6%
	Gamma Hurdle	0.098	94.0%	0.129	88.6%
	ZIC-BCF-Smear	0.046	99.3%	0.065	94.7%
DGP B: Gamma	DPglm	0.055	99.7%	0.077	98.2%
	Gamma Hurdle	0.098	94.0%	0.129	88.5%

Table 4: Hurdle-stage performance at the standard configuration (

N=500

, ZI Level 3,

\approx 40\%

zeros), across 100 seeds. RMSE is on the participation-probability scale and coverage is the empirical

95\%

credible-interval coverage of the true participation probability. BCF-Linear and Gamma + .01 do not model a hurdle stage.

Estimand	Model	Estimate	95% Credible Interval
Summed record duration (event-days)	ZIC-BCF-Smear	14.27	[4.69, 24.23]
Summed record duration (event-days)	Gamma Hurdle	15.95	[6.15, 26.30]
Distinct calendar days	ZIC-BCF-Smear	10.00	[3.57, 16.62]
Distinct calendar days	Gamma Hurdle	12.20	[5.56, 19.23]
Participation margin (probability)	ZIC-BCF-Smear	0.029	[ $-0.038$ , 0.102]
Participation margin (probability)	Gamma Hurdle	0.017	[ $-0.055$ , 0.089]

Table 5: Posterior average treatment effects of panitumumab assignment on severe-adverse-event burden, SPECTRUM analytic cohort (

N=520

). Positive values indicate a higher burden, or a higher probability of a positive reconstructed severe-adverse-event burden, under panitumumab. The hurdle margin is not literally the probability of any severe adverse event: 13 patients with severe events are reconstructed as zeros because their observation intervals were incomplete. The first four rows are on the event-day scale; the last two are differences in probability. The unadjusted arm differences, which randomization makes unbiased for the same estimand, are

15.68

and

12.06

event-days.

Window	Chemotherapy	Panitumumab	Difference
Last contact capped	19.18	30.90	11.73
Day 84, all patients	11.25	14.33	3.08
Day 84, observed to day 84	11.25	14.93	3.68
Day 120, all patients	15.55	19.58	4.03
Day 120, observed to day 120	16.12	21.25	5.13

Table 6: Mean accrued severe-adverse-event days by arm under common-horizon truncation and under capping at the patient-specific last-contact day (LASTCTDY), on the 520-person cohort. The observed-to-horizon rows restrict to patients whose last contact reaches the window (

n=213

chemotherapy and

219

panitumumab at day 84;

194

and

203

at day 120). Capping applies only to intervals with complete boundaries, so these are diagnostics bounding, not removing, the unresolved-record problem discussed above. The day-84 and day-120 rows apply the same last-contact cap, which is why they sit slightly below the plain uncapped truncation differences quoted in the text (

3.08

against

3.14

at day 84 and

4.03

against

4.10

at day 120).

Figure 1: ZIC-BCF-Smear subgroup conditional average treatment effects on summed severe-adverse-event-record duration, in event-days, with $95\%$ credible intervals; the dashed line marks no effect and the dotted line the overall effect of $14.3$ event-days. Every subgroup posterior mean falls within roughly five event-days of the overall effect and every interval covers it, so the estimated effect surface is broadly homogeneous. Interval width tracks subgroup size almost monotonically, widest for HPV-positive patients ( $n=69$ ), women ( $n=65$ ), and patients aged 70 or older ( $n=36$ ), which is the expected consequence of borrowing strength across a tree ensemble rather than fitting stratum-specific models.

		DGPs A & B (Hurdle)		DGP C (Tweedie)
Configuration	Target Zeros	Link-Linear $c$	Nonlinear $c$	Link-Linear $c$	Nonlinear $c$
Level 1 (Severe zero-inflation)	$\approx 85\%$	$-1.500$	$-1.367$	$-3.500$	$-5.648$
Level 2 (High zero-inflation)	$\approx 60\%$	$-0.500$	$-0.465$	$-2.000$	$-3.315$
Level 3 (Standard configuration)	$\approx 40\%$	$+0.200$	$+0.200$	$\phantom{+}0.000$	$-2.072$
Level 4 (Moderate zero-inflation)	$\approx 15\%$	$+1.000$	$+1.005$	$\phantom{+}0.000$	$-0.163$
Level 5 (Low zero-inflation)	$\approx 5\%$	$+1.800$	$+1.783$	$+1.000$	$+1.131$

Table 7: Intercept calibration parameter

c

and empirical zero proportions across the five zero-inflation sensitivity levels in the simulation studies.

	$\displaystyle\mu_{0}^{(s)}(X_{i})$	$\displaystyle=\Phi(\mu_{bi}^{(s)})\cdot\exp(\mu_{ci}^{(s)})\cdot\phi_{\text{smear}}^{(s)}$
	$\displaystyle\mu_{1}^{(s)}(X_{i})$	$\displaystyle=\Phi(\mu_{bi}^{(s)}+\tau_{bi}^{(s)})\cdot\exp(\mu_{ci}^{(s)}+\tau_{ci}^{(s)})\cdot\phi_{\text{smear}}^{(s)}$

Table 9

Data-generating process	Monitored set	Median $\widehat{R}$	95th pct. $\widehat{R}$	Max $\widehat{R}$	Median ESS	Min ESS	Max $\|z_{\mathrm{Geweke}}\|$
Linear DGP A (log-normal hurdle)	Global summaries	1.004	1.014	1.017	928	449	4.05
Linear DGP A (log-normal hurdle)	Unit-level CATEs	1.005	1.012	1.048	1,125	508	3.68
Linear DGP B (Gamma hurdle)	Global summaries	1.002	1.007	1.010	1,124	444	2.57
Linear DGP B (Gamma hurdle)	Unit-level CATEs	1.003	1.010	1.022	1,563	693	3.53
Linear DGP C (Tweedie)	Global summaries	1.004	1.038	1.118	1,113	203	4.08
Linear DGP C (Tweedie)	Unit-level CATEs	1.004	1.011	1.030	1,489	869	4.91

Table 8: Convergence diagnostics for the simulation grid: four chains of 1,000 retained draws on each of five replicates per data-generating process at

N=500

. Global summaries are the eight scalar functionals listed above; unit-level CATEs are ten units spanning the effect surface. The Geweke column is the largest absolute value over all chains and quantities in the cell.

Data-generating process	Median BP $p$	Rejected at $5\%$	Residual-scale ratio	Median relative change	Max $\|$ relative change $\|$
Linear DGP A (log-normal hurdle)	0.194	1 of 5	1.23	$+0.9\%$	$3.1\%$
Linear DGP B (Gamma hurdle)	0.026	3 of 5	1.21	$-4.4\%$	$8.5\%$
Linear DGP C (Tweedie)	$2.9\times 10^{-7}$	5 of 5	2.12	$-17.7\%$	$19.5\%$
Nonlinear DGP A (log-normal hurdle)	0.313	0 of 5	1.21	$+1.3\%$	$2.6\%$
Nonlinear DGP B (Gamma hurdle)	0.242	1 of 5	1.27	$+2.5\%$	$7.7\%$
Nonlinear DGP C (Tweedie)	$7.0\times 10^{-4}$	5 of 5	1.54	$-25.1\%$	$32.5\%$

Table 9: Homoskedasticity of the log-scale errors across the simulation grid, five replicates per data-generating process at

N=500

. The residual-scale ratio is the largest divided by the smallest residual standard deviation across five strata of the fitted log-scale mean. The relative change is the locally smeared average treatment effect divided by the scalar-smeared one, minus one.

Panel of Figure 2 — Figure 2: Full link-linear $N$ -sensitivity analysis (including the diverged Gamma + .01 benchmark) across DGP A, B, and C.

Panel of Figure 3 — Figure 3: Full link-linear zero-inflation sensitivity analysis (including the diverged Gamma + .01 benchmark) across DGP A, B, and C.

Panel of Figure 4 — Figure 4: Full link-nonlinear $N$ -sensitivity analysis under misspecification, displaying the extreme y-axis distortion caused by the diverged parametric benchmarks.

Panel of Figure 5 — Figure 5: Full link-nonlinear zero-inflation sensitivity analysis under misspecification, displaying the extreme y-axis distortion caused by the diverged parametric benchmarks.

Figure 6: Link-linear $N$ -sensitivity analysis for DGP A (Log-Normal Hurdle), excluding the diverged Gamma +.01 benchmark for legibility. Reading the four panels together, ZIC-BCF-Smear and the correctly specified Gamma Hurdle both converge as $N$ grows, but ZIC-BCF-Smear reaches the lower CATE RMSE with the shorter credible intervals (log scale, bottom right) while holding coverage at the nominal $95\%$ ; DPglm’s intervals remain an order of magnitude wider at every sample size, which is why its coverage is pinned at $100\%$ .

Figure 7: Link-linear zero-inflation sensitivity analysis for DGP A (Log-Normal Hurdle), excluding the diverged Gamma +.01 benchmark for legibility. RMSE is normalized by the true CATE scale so that cells with different zero proportions are comparable. The ZIC-BCF-Smear advantage over the single-part BCF-Linear baseline widens monotonically as the zero proportion rises toward $85\%$ , while the correctly specified Gamma Hurdle deteriorates at the right-hand end of the range as the active sample shrinks below $\approx 15\%$ of the data.

Panel of Figure 8 — Figure 8: Link-nonlinear $N$ -sensitivity analysis restricted to the numerically stable estimators (BCF-Linear, DPglm, and ZIC-BCF-Smear). Reading across the panels, ZIC-BCF-Smear holds the lowest CATE RMSE and the coverage closest to the nominal $95\%$ at every sample size in all three DGPs, while DPglm’s credible-interval length stays an order of magnitude wider throughout.

Panel of Figure 9 — Figure 9: Link-nonlinear zero-inflation sensitivity analysis restricted to the numerically stable estimators. The gap between ZIC-BCF-Smear and the single-part BCF-Linear baseline widens monotonically as the zero proportion rises toward $85\%$ , which is the graphical signature of fitting the intensity forest only on units that clear the hurdle.

DGP	Model	Diverged Seeds	CATE RMSE (median)	ATE Abs Err (median)	CATE Corr
DGP A: Log-Normal	Gamma Hurdle	4 / 100	3.960	0.759	-0.039
	Gamma +.01	5 / 100	4.336	0.747	-0.081
	ZIC-BCF-Smear	0 / 100	1.611	0.352	0.686
DGP B: Gamma	Gamma Hurdle	5 / 100	3.704	0.784	-0.040
	Gamma +.01	9 / 100	4.145	0.692	-0.068
	ZIC-BCF-Smear	0 / 100	1.537	0.494	0.645

Table 10: Numerical stability of the parametric benchmarks under nonlinear DGPs (

N=500

, ZI Level 3). Reports divergence rates and robust medians across 100 seeds.

DGP	Model	Parameter	$N=100$	$N=250$	$N=500$	$N=1000$	ZI 85% (Level 1)
DGP A: Log-Normal	DPglm	RMSE	0.086	0.067	0.053	0.057	0.050
	DPglm	Coverage	98.0%	99.5%	99.7%	99.7%	86.6%
	Gamma Hurdle	RMSE	0.208	0.129	0.098	0.071	0.080
	Gamma Hurdle	Coverage	95.1%	95.8%	94.0%	94.8%	95.1%
	ZIC-BCF-Smear	RMSE	0.068	0.054	0.046	0.040	0.036
	ZIC-BCF-Smear	Coverage	98.9%	99.0%	99.3%	98.6%	98.3%

Table 11: Linear study Hurdle stage sensitivity results across sample sizes

N

(under nominal zero-inflation) and zero-inflation levels (under

N=500

DGP	Model	Parameter	$N=100$	$N=250$	$N=500$	$N=1000$	ZI 85% (Level 1)
DGP A: Log-Normal	DPglm	RMSE	0.105	0.085	0.077	0.078	0.078
	DPglm	Coverage	98.2%	98.8%	98.6%	96.1%	96.9%
	Gamma Hurdle	RMSE	0.229	0.167	0.129	0.111	0.104
	Gamma Hurdle	Coverage	92.6%	89.9%	88.6%	82.2%	85.9%
	ZIC-BCF-Smear	RMSE	0.083	0.074	0.065	0.062	0.065
	ZIC-BCF-Smear	Coverage	98.0%	96.4%	94.7%	90.8%	92.8%

Table 12: Nonlinear study Hurdle stage sensitivity results across sample sizes

N

(under nominal zero-inflation) and zero-inflation levels (under

N=500

Arm	$n$	Zero (%)	Mean	Positive mean	Positive median	Maximum
Chemotherapy	260	24.6	26.75	35.49	20.5	263
Panitumumab + chemotherapy	260	22.7	42.43	54.89	36.0	641

Table 13: Observed summed-duration outcome by randomized arm. Positive-part summaries condition on

Y_{i}^{\mathrm{sum}}>0

Endpoint	Total days	Chemotherapy mean	Panitumumab + chemotherapy mean	Unadjusted difference	Zeros
Summed AE-record days	17,988	26.75	42.43	15.68	123
Distinct severe-AE calendar days	13,107	19.18	31.23	12.06	123

Table 14: Descriptive comparison of the summed-duration and distinct-calendar-day endpoints. The unadjusted difference is panitumumab plus chemotherapy minus chemotherapy.

Arm	$n$	Mean	SD	Median	Maximum
Chemotherapy	260	83.72	50.14	89.0	276
Panitumumab plus chemotherapy	260	130.30	127.19	112.5	1,149

Table 15: Treatment duration (TRTDUR, days) by randomized arm in the 520-person analytic cohort. TRTDUR is a post-randomization quantity and appears in no adjustment set.

Date variant and horizon	Chemotherapy	Panitumumab	Difference
Imputed start, imputed end (primary)
Uncapped	19.18	31.23	$+12.06$
120 days	15.55	19.65	$+4.10$
90 days	11.96	15.33	$+3.37$
84 days	11.25	14.39	$+3.14$
Raw start, raw end

Table 16: Mean distinct severe-adverse-event calendar days by arm under common-horizon truncation, on the 520-person basis with 260 patients per arm. The six patients with no adverse-event record contribute zero at every horizon. The primary row uses the imputed start and end days; the three remaining date variants recorded in the file are shown for comparison. This table is a diagnostic, not a corrected estimate; see the text.

Arm	Total treatment days	Total severe-AE days	Patient-level mean rate	Aggregate rate
Chemotherapy	21,766	4,986	53.72	22.91
Panitumumab plus chemotherapy	33,877	8,121	49.91	23.97

Table 17: Descriptive per-exposure burden by arm. The patient-level mean weights every patient equally and therefore up-weights patients with short exposure; the aggregate rate pools days over pooled treatment-time. Neither is an adjusted causal effect.

Arm	$n$	Completed safety follow-up	Mean DSDY	Mean LASTCTDY	LASTCTDY $<84$	LASTCTDY $\geq 84$
Chemotherapy	260	181 (69.6%)	110.7	376.8	47	213
Panitumumab plus chemotherapy	260	173 (66.5%)	159.5	403.5	41	219

Table 18: Disposition and observed follow-up records across randomized arms. DSDY denotes the recorded safety follow-up day, while LASTCTDY denotes the last-contact study day. Neither variable constitutes a fixed-horizon outcome by itself.

Model and estimand	Estimate	95% credible interval	$P(>0)$
ZIC-BCF-Smear, summed duration (event-days)	14.27	[4.69, 24.23]	0.999
Gamma Hurdle, summed duration (event-days)	15.95	[6.15, 26.30]	0.999
Unadjusted arm difference, summed duration	15.68	–	–
ZIC-BCF-Smear, distinct calendar days	10.00	[3.57, 16.62]	0.999
Gamma Hurdle, distinct calendar days	12.20	[5.56, 19.23]	1.000
Unadjusted arm difference, distinct calendar days	12.06	–	–

Table 19: Posterior average treatment effects. Positive values favor a higher burden or a higher probability of a positive reconstructed severe-AE burden under panitumumab assignment.

Figure 10: Posterior distributions of the average treatment effect on distinct severe-AE calendar days under ZIC-BCF-Smear (left) and the Gamma-hurdle benchmark (right). In each panel the solid vertical line marks the posterior mean, the dashed line the 2.5% posterior quantile, and the dotted lines the zero reference and the 97.5% posterior quantile.

Covariate	Level	$n$	CATE	95% credible interval	$P(>0)$
Overall	All	520	14.27	[4.69, 24.23]	0.999
Sex	Male	455	14.20	[4.53, 24.37]	0.999
Sex	Female	65	14.76	[2.28, 29.89]	0.990
ECOG status	ECOG 0	150	18.74	[5.76, 35.51]	1.000
ECOG status	ECOG 1	370	12.46	[3.27, 22.47]	0.996
Diagnosis site	Hypopharynx	75	13.37	[3.56, 24.47]	0.997

Table 20: ZIC-BCF-Smear subgroup CATEs for summed severe-AE-record duration.

Figure 11: ZIC-BCF-Smear subgroup CATEs for summed severe-AE-record duration, with 95% credible intervals. The dashed vertical line at zero is the no-effect reference and the dotted vertical line is the overall ATE of 14.3 event-days.

Figure 12: Gamma-hurdle subgroup CATEs for the same summed severe-AE-record duration outcome, on the same 95% credible-interval scale. The dotted vertical line is the Gamma-hurdle overall ATE of 16.1 event-days. Note that the horizontal axis spans roughly twice the range of Figure 11 .

Covariate	Level	$n$	CATE	95% credible interval	$P(>0)$
Overall	All	520	10.00	[3.57, 16.62]	0.999
Sex	Male	455	10.12	[3.55, 17.01]	0.999
Sex	Female	65	9.10	[0.40, 18.63]	0.980
ECOG status	ECOG 0	150	14.76	[4.86, 28.25]	1.000
ECOG status	ECOG 1	370	8.07	[1.85, 14.58]	0.994
Diagnosis site	Hypopharynx	75	9.55	[2.65, 17.22]	0.997

Table 21: ZIC-BCF-Smear subgroup CATEs for distinct severe-AE calendar days.

Figure 13: ZIC-BCF-Smear subgroup CATEs for distinct severe-AE calendar days, with 95% credible intervals. The dashed vertical line at zero is the no-effect reference and the dotted vertical line is the overall ATE of 10.1 days.

Figure 14: Decomposition of the dental-expenditure outcome into the zero point mass ( $\approx 53\%$ of records) and the positive continuous part.

Figure 15: Histogram of positive dental expenditures on the log scale, showing the strong right skew on the raw scale and the approximate log-normality that motivates the log-link continuous stage combined with a nonparametric (smearing) re-transformation.

Figure 16: Unadjusted probability of any dental expenditure by dental-insurance status. Coverage is strongly associated with clearing the zero hurdle; the causal analysis adjusts this association for confounding and separates it from the intensity effect.

Domain	$n$	Scalar CATE ($)	Locally smeared	Local 95% CrI	Change
Age group : 35-49	3468	175.27	183.95	[150.47, 218.70]	5.0%
Age group : 18-34	3352	169.06	176.81	[142.45, 214.46]	4.6%
Family income (weighted tertile) : High-income tertile	5330	188.25	196.54	[149.30, 242.36]	4.4%
Marital status (detailed) : Never married	4579	191.74	199.91	[160.15, 246.77]	4.3%
Region : Midwest	3837	195.04	202.46	[153.53, 259.09]	3.8%
Poverty cat. : High-income	7351	206.16	213.89	[164.10, 260.06]	3.7%

Table 22: Locally smeared subgroup sensitivity for the dollar-scale analytic-sample average: scalar Duan versus locally smeared subgroup CATEs, ordered by relative change. Both columns derive from the same pooled posterior draws; every locally smeared point estimate lies inside its scalar-smear 95% credible interval.

Figure 17: Posterior distribution of the survey-weighted ATE, with the design-aware $95\%$ interval marked. The posterior concentrates near $\$185$ and lies entirely above zero. — Figure 17: Posterior distribution of the survey-weighted ATE, with the design-aware $95\%$ interval marked. The posterior concentrates near $\$ 185$ and lies entirely above zero.

Figure 18: Marginal distribution of unit-level posterior-mean CATEs across all $18{,}763$ individuals. This marginal view is shown for reference only; all inferential subgroup summaries use the draw-by-draw construction of Section S6.2 .

Covariate	Subgroup	$N$	Analytic-sample average	Population target
Overall	All	18,763	188.02 [156.95, 220.67]	184.67 [154.63, 214.70]
Sex	Male	8,928	185.86 [153.06, 220.98]	182.56 [149.75, 215.38]
Sex	Female	9,835	189.99 [149.59, 230.49]	186.70 [148.40, 225.00]
Race and ethnicity	Hispanic	4,165	144.82 [113.97, 178.04]	146.57 [114.08, 179.06]
Race and ethnicity	White, non-Hispanic	10,243	218.16 [176.61, 260.39]	208.95 [170.75, 247.15]
Race and ethnicity	Black, non-Hispanic	2,495	138.37 [84.47, 192.74]	137.53 [84.85, 190.21]

Table 23: Subgroup CATE estimates (dollars), MEPS 2023. The sample column is the unweighted analytic-sample average with its

95\%

credible interval; the population column is the survey-weighted target with its design-aware

95\%

interval.

Figure 21: Posterior densities of the survey-weighted subgroup CATEs for race and ethnicity and for age. The heterogeneity is visible as a separation of whole posterior distributions, not merely of point estimates.

Figure 22: Unit-level posterior-mean CATE against age and family income, with a LOWESS trend. The insurance effect rises steadily with age and increases with income before flattening at the top of the distribution, consistent with the subgroup tables.

Covariate	Subgroup	$N$	Population target
Overall	All	18,763	0.139 [0.123, 0.155]
Sex	Male	8,928	0.143 [0.121, 0.165]
Sex	Female	9,835	0.136 [0.113, 0.158]
Race and ethnicity	Hispanic	4,165	0.129 [0.103, 0.155]
Race and ethnicity	White, non-Hispanic	10,243	0.151 [0.130, 0.171]
Race and ethnicity	Black, non-Hispanic	2,495	0.109 [0.063, 0.154]

Table 24: ZIC-BCF-Smear hurdle-margin (participation) subgroup CATEs, in probability points, on the survey-weighted population target with design-aware

95\%

intervals, MEPS 2023. Entries are the increase in the probability of any dental expenditure attributable to dental insurance.

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May 14, 2026

Causal Foundation Models with Continuous Treatments

Causal inference, estimating causal effects from observational data, is a fundamental tool in many disciplines. Of particular importance across a variety of domains is the continuous treatment setting, where the variable of intervention has a continuous range. This setting is far less explored and represents a substantial shift from the binary treatment setting, with models needing to represent effects across a continuum of treatment values. In this paper, we present the first causal foundation model for the continuous treatment setting. Our model meta-learns the ability to predict causal effects across a wide variety of unseen tasks without additional training or fine-tuning. First, we design a novel prior over data-generating processes with continuous treatment variables in order to generate a rich causal training corpus. We then train a transformer to reconstruct individual treatment-response curves given only observational data, leveraging in-context learning to amortize expensive Bayesian posterior inference. Our model achieves state-of-the-art performance on individual treatment-response curve reconstruction tasks compared to causal models which are trained specifically for those tasks. Inference code (including trained model weights) can be found at https://github.com/layer6ai-labs/CCPFN-inference .

Christopher Stith, Medha Barath, Vahid Balazadeh +2

Layer 6 AI · University of Toronto · Vector Institute

Causal Foundation Models Causal Effect Estimation Individualized Treatment Effect Estimation

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Apr 30, 2026

Bayesian X-Learner: Calibrated Posterior Inference for Heterogeneous Treatment Effects under Heavy-Tailed Outcomes

Conditional Average Treatment Effect (CATE) estimation in practice demands three properties simultaneously: heterogeneous effects

τ(x)

, calibrated uncertainty over them, and robustness to the heavy tails that contaminate real outcome data. Meta-learners (Künzel et al., 2019) give (i); causal forests and BART give (i)-(ii) with Gaussian-tail assumptions; no widely used tool gives all three. We present Bayesian X-Learner, an X-Learner built on cross-fitted doubly robust pseudo-outcomes (Kennedy, 2020) with a full MCMC posterior over

τ(x)

via a Welsch redescending pseudo-likelihood. On Hill's IHDP benchmark the default configuration attains mean

\sqrt{\varepsilon_{\mathrm{PEHE}}} = 0.56

on 5 replications (lowest mean; differences from S-/T-/X-learners, full-config Causal BART, and a causal forest baseline are not significant at

α=0.05

, and rank ordering is unstable at 10 replications -- IHDP comparisons are competitive rather than dominant). On contaminated "whale" DGPs with up to 20-25% tail density, a one-flag extension (contamination_severity) that selects a Huber-

δ

nuisance loss per Huber's minimax-

δ

relation recovers RMSE

\approx 0.13

with tight credible intervals (single-cross-fit 30-seed coverage 83% [Wilson 66%, 93%] at 20% density; modular-Bayes pooling with Bayesian-bootstrap nuisance draws restores nominal 95% coverage).

Eichi Uehara

Independent researcher

Robust Regression Causal Effect Estimation Conditional Average Treatment Effect Estimation Bayesian Inference Heavy-Tailed Distributions Individualized Treatment Effect Estimation+1

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Jul 4, 2026

Significance-First Splitting: Aligning Treatment Heterogeneity Detection with Honest Estimation

Estimating heterogeneous treatment effects (CATE) requires simultaneously detecting effect modification and quantifying estimation uncertainty. Existing tree-based methods make an uneasy trade-off: significance-based approaches (Radcliffe and Surry 2011) identify subgroup interactions directly but lack valid inference; honest causal trees (Athey and Imbens 2016) deliver nominal confidence interval coverage but use outcome-agnostic splitting criteria that sacrifice interaction sensitivity. We introduce a hybrid algorithm that fuses significance-based splitting with honest sample-splitting and cross-validation. Our splitting criterion uses the squared

t

-statistic for the treatment

\times

side interaction (

t^2

), which is shown to be directly aligned with the honest

\text{EMSE}_τ

criterion when the interaction is strong. Post-hoc honest cross-validation selects the cost-complexity penalty, giving a single principled estimator with nominal CI coverage at the leaf level. For forests, we retain bootstrap count vectors to enable an infinitesimal jackknife (IJ) variance estimate of Monte-Carlo convergence rather than formal pointwise inference. On the three synthetic designs from (Athey and Imbens 2016) the single tree achieves approximately 90% leaf-average CI coverage at the 90% nominal level across all three designs (200 replications each); on the Criteo, Hillstrom and Starbucks uplift datasets we match Qini coefficient performance of S-, T-learner and GRF baselines. An open-source Python package with reproducible seeds, sklearn-compatible API, and full test coverage accompanies this work (https://codeberg.org/hadjipantelis/rattus).

Pantelis Z. Hadjipantelis, Weng Man Chiang, Karthik Nagesh

TripAdvisor, UK

Causal Effect Estimation Confidence Region Estimation Uncertainty Quantification Conditional Average Treatment Effect Estimation Causal Inference Individualized Treatment Effect Estimation

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Abstract

Causal Effect Estimation

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