Table A.6: Tabular summary of Figure 4 including performance in other tasks as shown in Appendix A.7 but excluding performance when using the "Random" policy (see Table A.5 for those results). Here, we show the task accuracy of IntCEMs and CEMs when intervening with different test-time policies as we vary the fraction of concept groups we intervene on (indicated on the left-hand side of the dataset names).
Dataset	IntCEM (UCP)	CEM (UCP)	IntCEM (CooP)	CEM (CooP)	IntCEM (CVA)	CEM (CVA)	IntCEM (CVI)	CEM (CVI)	IntCEM (Skyline)	CEM (Skyline)
25%	MNIST-Add	94.07 ± 0.22	92.17 ± 0.46	93.16 ± 0.14	91.06 ± 0.41	93.14 ± 0.12	91.39 ± 0.65	95.14 ± 0.37	92.13 ± 0.70	99.93 ± 0.01	99.05 ± 0.26
MNIST-Add-Incomp	91.03 ± 0.38	88.53 ± 0.92	90.09 ± 0.34	87.63 ± 0.84	90.11 ± 0.30	88.21 ± 1.28	91.52 ± 0.68	88.37 ± 1.12	98.48 ± 0.08	95.06 ± 0.77
CUB	93.37 ± 0.20	88.56 ± 0.69	94.68 ± 0.25	89.09 ± 0.75	86.91 ± 1.05	84.81 ± 0.55	89.44 ± 1.03	86.01 ± 0.50	99.76 ± 0.05	95.25 ± 1.81
CUB-Incomp	82.20 ± 0.11	80.40 ± 0.26	82.34 ± 0.24	80.77 ± 0.47	79.49 ± 0.57	78.68 ± 0.51	79.08 ± 0.37	78.35 ± 0.53	90.16 ± 0.31	85.83 ± 0.72
CelebA	50.82 ± 0.41	38.67 ± 1.26	50.72 ± 0.56	38.83 ± 1.36	48.28 ± 0.34	36.04 ± 1.14	48.28 ± 0.34	36.04 ± 1.14	62.48 ± 0.54	44.34 ± 1.27
50%	MNIST-Add	96.60 ± 0.13	94.22 ± 0.55	94.74 ± 0.27	92.54 ± 0.44	95.70 ± 0.42	93.05 ± 0.83	96.97 ± 0.83	94.17 ± 0.78	100.00 ± 0.00	99.61 ± 0.15
MNIST-Add-Incomp	92.82 ± 0.45	89.97 ± 0.98	91.31 ± 0.59	88.89 ± 0.98	91.70 ± 0.46	89.19 ± 1.31	93.53 ± 0.64	89.59 ± 0.92	99.11 ± 0.18	96.17 ± 0.69
CUB	98.52 ± 0.19	93.70 ± 1.43	99.27 ± 0.16	93.60 ± 1.39	96.35 ± 0.70	91.71 ± 1.47	97.45 ± 0.40	91.49 ± 1.38	99.92 ± 0.01	96.87 ± 1.71
CUB-Incomp	91.21 ± 0.22	86.82 ± 0.73	91.25 ± 0.13	87.17 ± 0.74	88.31 ± 0.30	84.42 ± 0.59	87.90 ± 0.73	83.76 ± 1.07	96.38 ± 0.28	91.23 ± 0.73
CelebA	65.47 ± 0.59	46.29 ± 1.49	65.47 ± 0.57	46.48 ± 1.59	59.60 ± 0.38	42.61 ± 1.38	59.54 ± 0.39	42.61 ± 1.38	70.84 ± 0.47	48.56 ± 1.63
75%	MNIST-Add	98.78 ± 0.04	95.93 ± 0.65	97.99 ± 0.46	95.41 ± 0.66	97.80 ± 0.67	95.24 ± 0.88	98.43 ± 0.44	95.93 ± 0.69	100.00 ± 0.00	99.54 ± 0.19
MNIST-Add-Incomp	94.42 ± 0.22	91.06 ± 1.02	93.62 ± 0.82	90.83 ± 1.03	93.64 ± 0.15	90.43 ± 1.01	94.53 ± 0.23	90.74 ± 0.94	98.87 ± 0.15	95.81 ± 0.74
CUB	99.75 ± 0.07	95.93 ± 1.56	99.88 ± 0.04	95.90 ± 1.62	99.18 ± 0.23	95.17 ± 1.72	99.74 ± 0.04	95.22 ± 1.55	99.93 ± 0.02	96.96 ± 1.70
CUB-Incomp	95.04 ± 0.31	90.03 ± 0.87	95.37 ± 0.31	90.24 ± 0.79	92.92 ± 0.38	88.35 ± 0.85	93.81 ± 0.95	88.37 ± 1.36	96.68 ± 0.25	91.63 ± 0.69
CelebA	68.54 ± 0.67	47.43 ± 1.63	68.54 ± 0.66	47.51 ± 1.65	64.46 ± 0.47	45.65 ± 1.46	64.46 ± 0.47	45.65 ± 1.46	70.95 ± 0.52	48.61 ± 1.66
100%	MNIST-Add	99.51 ± 0.04	96.68 ± 0.68	99.51 ± 0.04	96.68 ± 0.68	99.51 ± 0.04	96.68 ± 0.68	99.51 ± 0.04	96.68 ± 0.68	99.51 ± 0.04	96.68 ± 0.68
MNIST-Add-Incomp	94.99 ± 0.11	91.53 ± 1.01	94.99 ± 0.11	91.53 ± 1.01	94.99 ± 0.11	91.53 ± 1.01	94.99 ± 0.11	91.53 ± 1.01	94.99 ± 0.11	91.53 ± 1.01
CUB	99.90 ± 0.04	96.75 ± 1.72	99.90 ± 0.04	96.75 ± 1.72	99.90 ± 0.04	96.75 ± 1.72	99.90 ± 0.04	96.75 ± 1.72	99.90 ± 0.04	96.75 ± 1.72
CUB-Incomp	96.52 ± 0.19	91.47 ± 0.67	96.52 ± 0.19	91.47 ± 0.67	96.52 ± 0.19	91.47 ± 0.67	96.52 ± 0.19	91.47 ± 0.67	96.52 ± 0.19	91.47 ± 0.67
CelebA	70.02 ± 0.62	48.10 ± 1.72	70.02 ± 0.62	48.10 ± 1.72	70.02 ± 0.62	48.10 ± 1.72	70.02 ± 0.62	48.10 ± 1.72	70.02 ± 0.62	48.10 ± 1.72