Learning Causal Normalizing Flows from Incomplete Data via Observed-Data Likelihood

Trung-Dung Hoang; Alceu Bissoto; Tim Flühmann; David Herzig; Christos Nakas; Lia Bally; Lisa M. Koch

cs.LGSep 29, 2026

Learning Causal Normalizing Flows from Incomplete Data via Observed-Data Likelihood

Authors: Trung-Dung Hoang, Alceu Bissoto, Tim Flühmann, David Herzig, Christos Nakas, Lia Bally, Lisa M. Koch

Organizations: Department of Digital Medicine, University of Bern, Switzerland · Department of Diabetes, Endocrinology, Nutritional Medicine and Metabolism (UDEM), Inselspital, Bern University Hospital, University of Bern, Switzerland · Graduate School for Cellular and Biomedical Sciences (GCB), University of Bern, Switzerland · Diabetes Center Berne, Switzerland · Department of Clinical Chemistry, Inselspital, Bern University Hospital, University of Bern, Switzerland · Laboratory of Biometry, School of Agriculture, University of Thessaly, 384 46 Volos, Greece

Abstract

Causal Normalizing Flows (CNFs) enable causal inference from observational data given the causal structure, but they assume fully observed training data. We introduce MissCNF, which trains CNFs directly on incomplete data by maximizing the marginal likelihood of each partially observed sample, without discarding rows or constructing a completed dataset. Thanks to the causal structure encoded in the autoregressive factorization of CNFs, only missing variables in the ancestral closure of the observed set are integrated out, while the others are dropped without computation. We further establish the conditions under which MissCNF recovers the true joint distribution, and introduce causal-family positivity, where identification is possible even when no record in the dataset is ever complete. We compare MissCNF with two common strategies for handling missing data: listwise deletion and impute-then-fit pipelines. Across eight synthetic causal benchmarks, three missingness mechanisms, and missing rates up to $90\%$ , MissCNF achieves the lowest KL divergence in 23 of 24 nonlinear MCAR and MAR settings and in all nonlinear MNAR settings, as well as the lowest counterfactual error in 20 of 24 settings. On linear SCMs, where linear imputation performs best, MissCNF ranks in the top two in 22 of 24 settings.

Figures & tables

Figure 1: (a) With

O=\{1,2,5\}

observed,

A

\{1,2,3,5\}

X_{3}

is a missing ancestor and must be integrated out.

X_{4}

is barren and is dropped without integration. (b) One MC sample of Algorithm 1 . Nodes in

A

are processed in topological order. Observed nodes are mapped forward to their noise and added to the log-likelihood. The missing

X_{3}

is generated from the noise

u_{3}^{(k)}

and passed on to its children.

X_{4}

is skipped.

Figure 2: Left: the two observed patterns. Right: the causal family

\{i\}\cup\mathrm{pa}(i)

of each node and the patterns that contain it. No pattern observes all of

V

, but every family is observed by at least one pattern.

Figure 3: Results under MCAR and MAR at 30%, 60%, and 90% missing rates (mean

\pm

std over 5 seeds; lower is better). (a) Symmetric KL divergence to the ground-truth distribution on all SCMs. (b)

\text{RMSE}_{\text{CF}}

on the nonlinear SCMs. The dashed line marks the model trained on complete data.

Figure 4: KL to the ground-truth distribution on Chain and Fork graphs (linear and nonlinear) under three regimes: Full positivity , Family positivity , and Family positivity violated . The dashed line marks the model trained on complete data.

Method	Chain ${}_{\text{LIN}}$	Fork ${}_{\text{LIN}}$	Collider ${}_{\text{LIN}}$	Triangle ${}_{\text{LIN}}$	Chain ${}_{\text{NLIN}}$	Fork ${}_{\text{NLIN}}$	Collider ${}_{\text{NLIN}}$	Triangle ${}_{\text{NLIN}}$
MICE	0.058 $\pm$ 0.046	0.030 $\pm$ 0.010	0.110 $\pm$ 0.080	0.014 $\pm$ 0.010	0.931 $\pm$ 0.279	19.116 $\pm$ 0.236	0.251 $\pm$ 0.146	16.060 $\pm$ 0.589
KPI	0.685 $\pm$ 0.220	0.997 $\pm$ 0.158	0.731 $\pm$ 0.349	0.650 $\pm$ 0.268	0.687 $\pm$ 0.301	1.574 $\pm$ 0.136	0.785 $\pm$ 0.399	5.596 $\pm$ 0.778
O-MIRACLE	3.808 $\pm$ 4.190	1.011 $\pm$ 0.386	0.956 $\pm$ 0.399	3.936 $\pm$ 4.829	2.749 $\pm$ 2.780	1.700 $\pm$ 0.423	0.921 $\pm$ 0.372	3.839 $\pm$ 2.393
MissCNF	0.083 $\pm$ 0.052	0.048 $\pm$ 0.025	0.117 $\pm$ 0.094	0.031 $\pm$ 0.016	0.121 $\pm$ 0.042	0.066 $\pm$ 0.030	0.177 $\pm$ 0.127	0.288 $\pm$ 0.280

Table 1: KL to the ground-truth distribution under self-masking MNAR at 60% missingness. Mean

\pm

std over 5 seeds. Lower is better. Bold = best, underline = second-best per task.

Figure 5: Scaling of MC samples and compute.

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 6 — Figure 6: Causal graph of the different SCMs considered in Section 4 ( Javaloy et al., 2023 ) .

MCAR, 30%
Method	Chain ${}_{\text{LIN}}$	Fork ${}_{\text{LIN}}$	Collider ${}_{\text{LIN}}$	Triangle ${}_{\text{LIN}}$	Chain ${}_{\text{NLIN}}$	Fork ${}_{\text{NLIN}}$	Collider ${}_{\text{NLIN}}$	Triangle ${}_{\text{NLIN}}$
Full data (reference)	0.005 $\pm$ 0.003	0.010 $\pm$ 0.004	0.003 $\pm$ 0.002	0.007 $\pm$ 0.005	0.002 $\pm$ 0.002	0.013 $\pm$ 0.003	0.004 $\pm$ 0.001	0.015 $\pm$ 0.007
Listwise deletion	0.011 $\pm$ 0.005	0.018 $\pm$ 0.006	0.003 $\pm$ 0.003	0.013 $\pm$ 0.007	0.005 $\pm$ 0.004	0.029 $\pm$ 0.004	0.004 $\pm$ 0.003	0.029 $\pm$ 0.014
Mean	13.642 $\pm$ 0.373	5.750 $\pm$ 0.176	0.158 $\pm$ 0.011	19.878 $\pm$ 0.948	0.963 $\pm$ 0.041	4.051 $\pm$ 0.132	0.078 $\pm$ 0.013	3.793 $\pm$ 0.071
MissForest	0.688 $\pm$ 0.075	0.582 $\pm$ 0.091	0.081 $\pm$ 0.008	0.266 $\pm$ 0.032	0.166 $\pm$ 0.010	1.830 $\pm$ 0.127	0.075 $\pm$ 0.013	0.845 $\pm$ 0.062
MICE	0.006 $\pm$ 0.005	0.009 $\pm$ 0.002	0.003 $\pm$ 0.002	0.010 $\pm$ 0.007	0.177 $\pm$ 0.018	9.056 $\pm$ 0.181	0.016 $\pm$ 0.002	8.357 $\pm$ 0.287

Appendix

Table 2: KL across all 8 SCMs, both missingness mechanisms and three missingness rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (mechanism

\times

rate); the full-data reference row (model trained with no missingness) is excluded from best/second-best marking.

MCAR, 30%
Method	Chain ${}_{\text{LIN}}$	Fork ${}_{\text{LIN}}$	Collider ${}_{\text{LIN}}$	Triangle ${}_{\text{LIN}}$	Chain ${}_{\text{NLIN}}$	Fork ${}_{\text{NLIN}}$	Collider ${}_{\text{NLIN}}$	Triangle ${}_{\text{NLIN}}$
Full data (reference)	0.054 $\pm$ 0.014	0.033 $\pm$ 0.004	0.011 $\pm$ 0.001	0.350 $\pm$ 0.074	0.027 $\pm$ 0.008	0.063 $\pm$ 0.014	0.018 $\pm$ 0.006	0.145 $\pm$ 0.048
Listwise deletion	0.089 $\pm$ 0.035	0.035 $\pm$ 0.006	0.013 $\pm$ 0.002	0.333 $\pm$ 0.052	0.040 $\pm$ 0.011	0.083 $\pm$ 0.030	0.019 $\pm$ 0.007	0.219 $\pm$ 0.091
Mean	1.778 $\pm$ 0.043	0.299 $\pm$ 0.010	0.067 $\pm$ 0.004	3.809 $\pm$ 0.311	0.599 $\pm$ 0.016	0.772 $\pm$ 0.048	0.027 $\pm$ 0.006	1.354 $\pm$ 0.189
MissForest	0.407 $\pm$ 0.090	0.057 $\pm$ 0.012	0.012 $\pm$ 0.002	0.373 $\pm$ 0.289	0.077 $\pm$ 0.021	0.580 $\pm$ 0.172	0.028 $\pm$ 0.009	1.130 $\pm$ 0.473
MICE	0.070 $\pm$ 0.008	0.032 $\pm$ 0.004	0.012 $\pm$ 0.003	0.372 $\pm$ 0.046	0.122 $\pm$ 0.005	1.187 $\pm$ 0.037	0.026 $\pm$ 0.007	2.184 $\pm$ 0.062

Appendix

Table 3: RMSE

{}_{\text{ATE}}

across all 8 SCMs, both missingness mechanisms and three missingness rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (mechanism

\times

rate); the full-data reference row (model trained with no missingness) is excluded from best/second-best marking.

MCAR, 30%
Method	Chain ${}_{\text{LIN}}$	Fork ${}_{\text{LIN}}$	Collider ${}_{\text{LIN}}$	Triangle ${}_{\text{LIN}}$	Chain ${}_{\text{NLIN}}$	Fork ${}_{\text{NLIN}}$	Collider ${}_{\text{NLIN}}$	Triangle ${}_{\text{NLIN}}$
Full data (reference)	0.067 $\pm$ 0.012	0.022 $\pm$ 0.002	0.016 $\pm$ 0.001	0.268 $\pm$ 0.037	0.036 $\pm$ 0.002	0.101 $\pm$ 0.011	0.034 $\pm$ 0.002	0.198 $\pm$ 0.037
Listwise deletion	0.091 $\pm$ 0.017	0.028 $\pm$ 0.003	0.019 $\pm$ 0.001	0.287 $\pm$ 0.036	0.049 $\pm$ 0.006	0.137 $\pm$ 0.013	0.038 $\pm$ 0.005	0.287 $\pm$ 0.053
Mean	1.372 $\pm$ 0.037	0.218 $\pm$ 0.011	0.066 $\pm$ 0.003	3.381 $\pm$ 0.242	0.500 $\pm$ 0.011	1.082 $\pm$ 0.021	0.071 $\pm$ 0.008	1.825 $\pm$ 0.064
MissForest	0.390 $\pm$ 0.044	0.073 $\pm$ 0.006	0.029 $\pm$ 0.005	0.456 $\pm$ 0.202	0.121 $\pm$ 0.014	0.971 $\pm$ 0.086	0.050 $\pm$ 0.009	1.381 $\pm$ 0.307
MICE	0.075 $\pm$ 0.010	0.022 $\pm$ 0.001	0.018 $\pm$ 0.003	0.283 $\pm$ 0.027	0.140 $\pm$ 0.004	1.526 $\pm$ 0.038	0.063 $\pm$ 0.010	2.178 $\pm$ 0.042

Appendix

Table 4: RMSE

{}_{\text{CF}}

across all 8 SCMs, both missingness mechanisms and three missingness rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (mechanism

\times

rate); the full-data reference row (model trained with no missingness) is excluded from best/second-best marking.

Task	Setting	MissCNF	MICE	O-MIRACLE	KPI	DiffPuter
Chain ${}_{\text{LIN}}$	Full pos.	0.023 $\pm$ 0.010	0.007 $\pm$ 0.005	1.27 $\pm$ 1.13	0.345 $\pm$ 0.102	5.20 $\pm$ 3.81
	Cov. hold	0.068 $\pm$ 0.023	0.007 $\pm$ 0.005	2.70 $\pm$ 0.113	0.338 $\pm$ 0.026	14.4 $\pm$ 0.438
	Cov. violated	0.385 $\pm$ 0.116	0.005 $\pm$ 0.004	439 $\pm$ 277	1309 $\pm$ 1082	1434 $\pm$ 116
Chain ${}_{\text{NLIN}}$	Full pos.	0.023 $\pm$ 0.026	0.610 $\pm$ 0.213	0.759 $\pm$ 0.284	0.290 $\pm$ 0.129	1.30 $\pm$ 0.688
	Cov. hold	0.024 $\pm$ 0.014	0.296 $\pm$ 0.021	1.63 $\pm$ 0.381	0.345 $\pm$ 0.047	4.64 $\pm$ 4.71
	Cov. violated	0.034 $\pm$ 0.017	0.703 $\pm$ 0.042	15.9 $\pm$ 0.530	14.0 $\pm$ 2.70	19.0 $\pm$ 1.61

Appendix

Table 5: KL across all 4 SCMs under 3 settings. Values are mean

\pm

std over seeds. Lower is better. Bold = best, within each row.

Task	Setting	MissCNF	MICE	O-MIRACLE	KPI	DiffPuter
Chain ${}_{\text{LIN}}$	Full pos.	0.078 $\pm$ 0.035	0.072 $\pm$ 0.021	0.083 $\pm$ 0.029	0.102 $\pm$ 0.036	0.504 $\pm$ 0.234
	Cov. hold	0.117 $\pm$ 0.055	0.057 $\pm$ 0.030	0.111 $\pm$ 0.044	0.148 $\pm$ 0.065	1.44 $\pm$ 0.196
	Cov. violated	0.157 $\pm$ 0.057	0.064 $\pm$ 0.026	2.01 $\pm$ 0.096	29.1 $\pm$ 43.7	3.51 $\pm$ 0.248
Chain ${}_{\text{NLIN}}$	Full pos.	0.041 $\pm$ 0.011	0.069 $\pm$ 0.007	0.034 $\pm$ 0.006	0.134 $\pm$ 0.019	0.165 $\pm$ 0.197
	Cov. hold	0.041 $\pm$ 0.011	0.062 $\pm$ 0.013	0.067 $\pm$ 0.014	0.079 $\pm$ 0.005	0.537 $\pm$ 0.017
	Cov. violated	0.042 $\pm$ 0.018	0.176 $\pm$ 0.017	0.470 $\pm$ 0.021	0.468 $\pm$ 0.020	0.985 $\pm$ 0.028

Appendix

Table 6: RMSE

{}_{\text{ATE}}

across all 4 SCMs under 3 settings. Values are mean

\pm

std over seeds. Lower is better. Bold = best, within each row.

Task	Setting	MissCNF	MICE	O-MIRACLE	KPI	DiffPuter
Chain ${}_{\text{LIN}}$	Full pos.	0.104 $\pm$ 0.006	0.075 $\pm$ 0.007	0.264 $\pm$ 0.058	0.166 $\pm$ 0.028	0.569 $\pm$ 0.129
	Cov. hold	0.133 $\pm$ 0.029	0.068 $\pm$ 0.010	0.472 $\pm$ 0.009	0.277 $\pm$ 0.014	1.40 $\pm$ 0.071
	Cov. violated	0.200 $\pm$ 0.030	0.076 $\pm$ 0.011	5.69 $\pm$ 1.62	33.4 $\pm$ 18.8	92.3 $\pm$ 71.5
Chain ${}_{\text{NLIN}}$	Full pos.	0.057 $\pm$ 0.004	0.207 $\pm$ 0.017	0.163 $\pm$ 0.033	0.163 $\pm$ 0.008	0.267 $\pm$ 0.129
	Cov. hold	0.057 $\pm$ 0.010	0.204 $\pm$ 0.016	0.258 $\pm$ 0.012	0.159 $\pm$ 0.012	0.707 $\pm$ 0.086
	Cov. violated	0.058 $\pm$ 0.011	0.296 $\pm$ 0.033	0.965 $\pm$ 0.034	0.813 $\pm$ 0.034	1.52 $\pm$ 0.135

Appendix

Table 7: RMSE

{}_{\text{CF}}

across all 4 SCMs under 3 settings. Values are mean

\pm

std over seeds. Lower is better. Bold = best, within each row.

Figure 7: Local KL error (defined in Appendix B.5.4 ) as a function of the distance to the identification boundary. The dashed line denotes the boundary

x_{\mathrm{rel}}=0

. Negative values correspond to the revealed side, while positive values lie inside the non-identified region

Z_{o}

. Curves show mean

\pm

std over seeds.

MNAR (self-masking), 30%
Method	Chain ${}_{\text{LIN}}$	Fork ${}_{\text{LIN}}$	Collider ${}_{\text{LIN}}$	Triangle ${}_{\text{LIN}}$	Chain ${}_{\text{NLIN}}$	Fork ${}_{\text{NLIN}}$	Collider ${}_{\text{NLIN}}$	Triangle ${}_{\text{NLIN}}$
MICE	0.020 $\pm$ 0.012	0.013 $\pm$ 0.004	0.035 $\pm$ 0.026	0.008 $\pm$ 0.006	0.446 $\pm$ 0.204	9.367 $\pm$ 0.115	0.087 $\pm$ 0.050	9.131 $\pm$ 0.391
KPI	0.132 $\pm$ 0.026	0.383 $\pm$ 0.031	0.131 $\pm$ 0.056	0.121 $\pm$ 0.024	0.261 $\pm$ 0.149	0.678 $\pm$ 0.047	0.149 $\pm$ 0.070	2.951 $\pm$ 0.308
O-MIRACLE	0.210 $\pm$ 0.030	0.204 $\pm$ 0.063	0.152 $\pm$ 0.062	0.168 $\pm$ 0.025	0.312 $\pm$ 0.167	0.426 $\pm$ 0.070	0.171 $\pm$ 0.075	0.921 $\pm$ 0.212
MissCNF (ours)	0.028 $\pm$ 0.014	0.021 $\pm$ 0.014	0.037 $\pm$ 0.025	0.014 $\pm$ 0.008	0.054 $\pm$ 0.033	0.022 $\pm$ 0.005	0.060 $\pm$ 0.040	0.078 $\pm$ 0.043
MNAR (self-masking), 60%

Appendix

Table 8: KL across all 8 SCMs under self-masking MNAR at two missing rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (missing rate).

MNAR (self-masking), 30%
Method	Chain ${}_{\text{LIN}}$	Fork ${}_{\text{LIN}}$	Collider ${}_{\text{LIN}}$	Triangle ${}_{\text{LIN}}$	Chain ${}_{\text{NLIN}}$	Fork ${}_{\text{NLIN}}$	Collider ${}_{\text{NLIN}}$	Triangle ${}_{\text{NLIN}}$
MICE	0.086 $\pm$ 0.033	0.034 $\pm$ 0.003	0.014 $\pm$ 0.006	0.312 $\pm$ 0.076	0.077 $\pm$ 0.040	1.385 $\pm$ 0.032	0.029 $\pm$ 0.006	1.698 $\pm$ 0.101
KPI	0.100 $\pm$ 0.045	0.046 $\pm$ 0.005	0.015 $\pm$ 0.005	0.872 $\pm$ 0.058	0.113 $\pm$ 0.014	0.192 $\pm$ 0.081	0.024 $\pm$ 0.008	1.304 $\pm$ 0.274
O-MIRACLE	0.083 $\pm$ 0.045	0.035 $\pm$ 0.004	0.014 $\pm$ 0.006	0.349 $\pm$ 0.097	0.054 $\pm$ 0.024	0.163 $\pm$ 0.062	0.025 $\pm$ 0.004	0.477 $\pm$ 0.116
MissCNF (ours)	0.086 $\pm$ 0.037	0.035 $\pm$ 0.004	0.015 $\pm$ 0.006	0.359 $\pm$ 0.117	0.049 $\pm$ 0.015	0.074 $\pm$ 0.032	0.025 $\pm$ 0.012	0.216 $\pm$ 0.092
MNAR (self-masking), 60%

Appendix

Table 9: RMSE

{}_{\text{ATE}}

across all 8 SCMs under self-masking MNAR at two missing rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (missing rate).

MNAR (self-masking), 30%
Method	Chain ${}_{\text{LIN}}$	Fork ${}_{\text{LIN}}$	Collider ${}_{\text{LIN}}$	Triangle ${}_{\text{LIN}}$	Chain ${}_{\text{NLIN}}$	Fork ${}_{\text{NLIN}}$	Collider ${}_{\text{NLIN}}$	Triangle ${}_{\text{NLIN}}$
MICE	0.090 $\pm$ 0.033	0.022 $\pm$ 0.001	0.019 $\pm$ 0.004	0.268 $\pm$ 0.053	0.149 $\pm$ 0.009	1.654 $\pm$ 0.027	0.091 $\pm$ 0.015	1.928 $\pm$ 0.057
KPI	0.129 $\pm$ 0.040	0.039 $\pm$ 0.001	0.031 $\pm$ 0.007	0.906 $\pm$ 0.082	0.132 $\pm$ 0.008	0.414 $\pm$ 0.037	0.066 $\pm$ 0.020	1.634 $\pm$ 0.078
O-MIRACLE	0.146 $\pm$ 0.048	0.029 $\pm$ 0.004	0.025 $\pm$ 0.004	0.330 $\pm$ 0.056	0.104 $\pm$ 0.019	0.325 $\pm$ 0.031	0.053 $\pm$ 0.011	0.920 $\pm$ 0.190
MissCNF (ours)	0.101 $\pm$ 0.034	0.025 $\pm$ 0.001	0.023 $\pm$ 0.003	0.328 $\pm$ 0.096	0.059 $\pm$ 0.003	0.113 $\pm$ 0.013	0.045 $\pm$ 0.009	0.317 $\pm$ 0.086
MNAR (self-masking), 60%

Appendix

Table 10: RMSE

{}_{\text{CF}}

across all 8 SCMs under self-masking MNAR at two missing rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (missing rate).

	Fork ${}_{\text{NLIN}}$				Triangle ${}_{\text{NLIN}}$
$K$	KL	RMSE ${}_{\text{ATE}}$	RMSE ${}_{\text{CF}}$	Time (s)	KL	RMSE ${}_{\text{ATE}}$	RMSE ${}_{\text{CF}}$	Time (s)
32	0.054 $\pm$ 0.015	0.078 $\pm$ 0.026	0.170 $\pm$ 0.012	272 $\pm$ 3	0.086 $\pm$ 0.028	0.248 $\pm$ 0.118	0.433 $\pm$ 0.102	272 $\pm$ 6
64	0.041 $\pm$ 0.015	0.085 $\pm$ 0.028	0.157 $\pm$ 0.006	273 $\pm$ 4	0.066 $\pm$ 0.023	0.190 $\pm$ 0.086	0.343 $\pm$ 0.073	270 $\pm$ 4
128	0.036 $\pm$ 0.015	0.083 $\pm$ 0.024	0.154 $\pm$ 0.006	284 $\pm$ 4	0.055 $\pm$ 0.022	0.251 $\pm$ 0.093	0.374 $\pm$ 0.116	284 $\pm$ 4
256	0.032 $\pm$ 0.012	0.076 $\pm$ 0.024	0.149 $\pm$ 0.005	328 $\pm$ 4	0.056 $\pm$ 0.023	0.193 $\pm$ 0.053	0.322 $\pm$ 0.019	329 $\pm$ 4
512	0.032 $\pm$ 0.011	0.083 $\pm$ 0.024	0.147 $\pm$ 0.004	416 $\pm$ 4	0.052 $\pm$ 0.032	0.190 $\pm$ 0.043	0.325 $\pm$ 0.055	426 $\pm$ 13

Appendix

Table 11: Monte Carlo ablation on Fork

{}_{\text{NLIN}}

and Triangle

{}_{\text{NLIN}}

K

is the number of Monte Carlo samples used to estimate the observed-data marginal likelihood. Values are mean

\pm

std over seeds. Time is total training time in seconds.

Figure 8: KL to the ground-truth distribution as

\alpha

increases, on

\text{Chain}_{\alpha}

under MCAR and MAR at 30% and 60% missing rates (mean ± std over seeds).

\alpha

interpolates the

X_{2}\rightarrow X_{3}

mechanism from purely linear (

\alpha=0

) to purely quadratic (

\alpha=1

MCAR, 30%
Method	$\alpha=0.00$	$\alpha=0.25$	$\alpha=0.50$	$\alpha=0.75$	$\alpha=1.00$
Listwise deletion	0.006 $\pm$ 0.002	0.006 $\pm$ 0.001	0.008 $\pm$ 0.002	0.008 $\pm$ 0.003	0.008 $\pm$ 0.001
MICE	0.004 $\pm$ 0.001	0.029 $\pm$ 0.004	0.151 $\pm$ 0.013	0.312 $\pm$ 0.021	0.345 $\pm$ 0.021
KPI	0.118 $\pm$ 0.012	0.128 $\pm$ 0.014	0.146 $\pm$ 0.014	0.140 $\pm$ 0.014	0.137 $\pm$ 0.016
DiffPuter	0.169 $\pm$ 0.007	0.156 $\pm$ 0.006	0.129 $\pm$ 0.020	0.119 $\pm$ 0.015	0.104 $\pm$ 0.012
O-MIRACLE	0.082 $\pm$ 0.004	0.084 $\pm$ 0.006	0.095 $\pm$ 0.011	0.097 $\pm$ 0.011	0.099 $\pm$ 0.008

Appendix

Table 12: KL vs. nonlinearity strength

\alpha

on the

\text{Chain}_{\alpha}

graph, both missingness mechanisms and two missing rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (mechanism

\times

rate).

MCAR, 30%
Method	$\alpha=0.00$	$\alpha=0.25$	$\alpha=0.50$	$\alpha=0.75$	$\alpha=1.00$
Listwise deletion	0.027 $\pm$ 0.006	0.027 $\pm$ 0.008	0.025 $\pm$ 0.011	0.022 $\pm$ 0.009	0.023 $\pm$ 0.006
MICE	0.021 $\pm$ 0.006	0.020 $\pm$ 0.003	0.030 $\pm$ 0.007	0.041 $\pm$ 0.006	0.048 $\pm$ 0.007
KPI	0.086 $\pm$ 0.016	0.083 $\pm$ 0.011	0.064 $\pm$ 0.006	0.038 $\pm$ 0.008	0.029 $\pm$ 0.008
DiffPuter	0.039 $\pm$ 0.013	0.038 $\pm$ 0.011	0.042 $\pm$ 0.015	0.038 $\pm$ 0.009	0.037 $\pm$ 0.006
O-MIRACLE	0.021 $\pm$ 0.005	0.020 $\pm$ 0.005	0.019 $\pm$ 0.005	0.018 $\pm$ 0.005	0.019 $\pm$ 0.003

Appendix

Table 13: RMSE

{}_{\text{ATE}}

vs. nonlinearity strength

\alpha

on the

\text{Chain}_{\alpha}

graph, both missingness mechanisms and two missing rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (mechanism

\times

rate).

MCAR, 30%
Method	$\alpha=0.00$	$\alpha=0.25$	$\alpha=0.50$	$\alpha=0.75$	$\alpha=1.00$
Listwise deletion	0.024 $\pm$ 0.002	0.026 $\pm$ 0.002	0.027 $\pm$ 0.004	0.027 $\pm$ 0.004	0.026 $\pm$ 0.003
MICE	0.020 $\pm$ 0.003	0.041 $\pm$ 0.001	0.085 $\pm$ 0.002	0.119 $\pm$ 0.002	0.124 $\pm$ 0.002
KPI	0.085 $\pm$ 0.009	0.083 $\pm$ 0.005	0.083 $\pm$ 0.002	0.080 $\pm$ 0.003	0.077 $\pm$ 0.006
DiffPuter	0.041 $\pm$ 0.010	0.042 $\pm$ 0.007	0.046 $\pm$ 0.003	0.042 $\pm$ 0.007	0.041 $\pm$ 0.006
O-MIRACLE	0.023 $\pm$ 0.001	0.029 $\pm$ 0.002	0.038 $\pm$ 0.003	0.033 $\pm$ 0.002	0.034 $\pm$ 0.006

Appendix

Table 14: RMSE

{}_{\text{CF}}

vs. nonlinearity strength

\alpha

on the

\text{Chain}_{\alpha}

graph, both missingness mechanisms and two missing rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (mechanism

\times

rate).

MCAR, 30%
Method	$\alpha=0.00$	$\alpha=0.25$	$\alpha=0.50$	$\alpha=0.75$	$\alpha=1.00$
Listwise deletion	0.004 $\pm$ 0.004	0.007 $\pm$ 0.003	0.009 $\pm$ 0.002	0.010 $\pm$ 0.001	0.008 $\pm$ 0.002
MICE	0.003 $\pm$ 0.003	0.029 $\pm$ 0.004	0.168 $\pm$ 0.010	0.349 $\pm$ 0.014	0.398 $\pm$ 0.016
KPI	0.055 $\pm$ 0.010	0.056 $\pm$ 0.012	0.059 $\pm$ 0.010	0.061 $\pm$ 0.007	0.065 $\pm$ 0.010
DiffPuter	0.057 $\pm$ 0.007	0.060 $\pm$ 0.011	0.068 $\pm$ 0.008	0.068 $\pm$ 0.010	837.056 $\pm$ 1871.559
O-MIRACLE	0.071 $\pm$ 0.010	0.073 $\pm$ 0.011	0.075 $\pm$ 0.010	0.079 $\pm$ 0.011	0.079 $\pm$ 0.008

Appendix

Table 15: KL vs. nonlinearity strength

\alpha

on the

\text{Collider}_{\alpha}

graph, both missingness mechanisms and two missing rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (mechanism

\times

rate).

MCAR, 30%
Method	$\alpha=0.00$	$\alpha=0.25$	$\alpha=0.50$	$\alpha=0.75$	$\alpha=1.00$
Listwise deletion	0.017 $\pm$ 0.005	0.015 $\pm$ 0.004	0.015 $\pm$ 0.003	0.015 $\pm$ 0.003	0.015 $\pm$ 0.003
MICE	0.016 $\pm$ 0.003	0.015 $\pm$ 0.001	0.020 $\pm$ 0.003	0.022 $\pm$ 0.005	0.026 $\pm$ 0.003
KPI	0.015 $\pm$ 0.005	0.014 $\pm$ 0.005	0.014 $\pm$ 0.005	0.016 $\pm$ 0.004	0.014 $\pm$ 0.004
DiffPuter	0.017 $\pm$ 0.005	0.015 $\pm$ 0.006	0.015 $\pm$ 0.006	0.020 $\pm$ 0.009	0.531 $\pm$ 1.153
O-MIRACLE	0.015 $\pm$ 0.003	0.013 $\pm$ 0.003	0.013 $\pm$ 0.003	0.013 $\pm$ 0.004	0.013 $\pm$ 0.003

Appendix

Table 16: RMSE

{}_{\text{ATE}}

vs. nonlinearity strength

\alpha

on the

\text{Collider}_{\alpha}

graph, both missingness mechanisms and two missing rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (mechanism

\times

rate).

MCAR, 30%
Method	$\alpha=0.00$	$\alpha=0.25$	$\alpha=0.50$	$\alpha=0.75$	$\alpha=1.00$
Listwise deletion	0.020 $\pm$ 0.003	0.025 $\pm$ 0.003	0.028 $\pm$ 0.002	0.029 $\pm$ 0.002	0.029 $\pm$ 0.002
MICE	0.019 $\pm$ 0.002	0.033 $\pm$ 0.004	0.064 $\pm$ 0.005	0.080 $\pm$ 0.005	0.086 $\pm$ 0.003
KPI	0.026 $\pm$ 0.004	0.028 $\pm$ 0.004	0.030 $\pm$ 0.004	0.031 $\pm$ 0.001	0.031 $\pm$ 0.006
DiffPuter	0.021 $\pm$ 0.003	0.024 $\pm$ 0.001	0.028 $\pm$ 0.004	0.031 $\pm$ 0.003	0.422 $\pm$ 0.873
O-MIRACLE	0.020 $\pm$ 0.002	0.023 $\pm$ 0.003	0.025 $\pm$ 0.002	0.026 $\pm$ 0.003	0.026 $\pm$ 0.003

Appendix

Table 17: RMSE

{}_{\text{CF}}

vs. nonlinearity strength

\alpha

on the

\text{Collider}_{\alpha}

graph, both missingness mechanisms and two missing rates. Values are mean

\pm

std over seeds. Lower is better. Bold = best, underline = second-best, within each row-block (mechanism

\times

rate).

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Generalizing Identifiability Results in Causal Discovery from Complete Data to Missing Data Settings

Matteo Ceriscioli, Shohei Shimizu, Mallika Gupta, Karthika Mohan

Oregon State University, Causal Intelligence and Reasoning Lab, Corvallis, USA · RIKEN Advanced Intelligence Project, Causal Inference Team, Tokyo, Japan · The University of Osaka, Reasoning for Intelligence Department, Osaka, Japan+1

Causal Discovery Causal Identifiability Learning with Missing Data

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While causal discovery methods have matured for complete data, such algorithms for handling missing values remain limited. We establish general sufficient conditions under which causal graphs remain identifiable (i.e., uniquely determined from the available data distribution) under various missingness categories, including Missing Not At Random, and provide a framework to extend correctness guarantees from causal discovery algorithms for complete data to missing data settings. Our results show that identifiability of causal graphs with no unobserved confounders is preserved when (1) missingness mechanisms do not causally interact, (2) no variable directly causes its own missingness, and (3) either faithfulness holds for missingness mechanisms, or they follow generalized linear models. Guided by these theoretical results, we introduce two asymptotically correct algorithms for Additive Noise Models (ANMs) that recover the underlying causal graph and provide consistent parameter estimates.

Abstract

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