Table A.5: Tabular summary of Figure 3. Task predictive performance (accuracy or AUC in %) across all tasks and baselines when intervening on a fixed fraction of the set of available concepts at test time (indicated on the left-hand side of the dataset names). Except for "IntCEM (Learnt Policy  $\psi$ )", all interventions use a random intervention policy. We write in bold the best performance results across baselines following a random intervention policy.
Dataset	IntCEM (Random)	CEM	Joint CBM-Sigmoid	Joint CBM-Logit	Ind. CBM	Seq. CBM	IntCEM (Learnt Policy \( \psi \) )
25%	MNIST-Add	\( {93.78} \pm {0.09} \)	\( {91.66} \pm {0.41} \)	\( {86.47} \pm {1.11} \)	\( {79.86} \pm {3.12} \)	\( {84.11} \pm {1.09} \)	\( {88.88} \pm {0.67} \)	\( {94.75} \pm {0.28} \)
MNIST-Add-Incomp	\( {90.88} \pm {0.25} \)	\( {88.22} \pm {0.92} \)	\( {84.68} \pm {0.83} \)	\( {77.59} \pm {1.57} \)	\( {86.06} \pm {0.77} \)	\( {87.53} \pm {0.74} \)	\( {91.69} \pm {0.39} \)
CUB	\( {88.53} \pm {0.23} \)	\( {84.08} \pm {0.36} \)	\( {83.18} \pm {0.22} \)	\( {81.76} \pm {0.90} \)	\( {79.96} \pm {1.57} \)	\( {65.40} \pm {5.38} \)	\( {94.10} \pm {0.49} \)
CUB-Incomp	\( {78.34} \pm {0.34} \)	\( {77.83} \pm {0.33} \)	\( {60.94} \pm {3.66} \)	\( {62.97} \pm {7.19} \)	\( {48.16} \pm {0.62} \)	\( {42.78} \pm {1.63} \)	\( {81.71} \pm {0.23} \)
CelebA	\( {42.25} \pm {0.36} \)	\( {33.27} \pm {0.94} \)	\( {25.16} \pm {0.58} \)	\( {20.02} \pm {1.27} \)	\( {25.63} \pm {0.55} \)	\( {26.21} \pm {0.32} \)	\( {47.63} \pm {1.49} \)
50%	MNIST-Add	\( {95.70} \pm {0.11} \)	\( {93.36} \pm {0.51} \)	\( {88.66} \pm {1.20} \)	\( {80.69} \pm {3.30} \)	\( {85.71} \pm {0.94} \)	\( {90.87} \pm {0.83} \)	\( {96.71} \pm {0.20} \)
MNIST-Add-Incomp	\( {92.07} \pm {0.21} \)	\( {89.13} \pm {0.96} \)	\( {85.73} \pm {1.12} \)	\( {77.96} \pm {1.73} \)	\( {86.87} \pm {0.74} \)	\( {88.58} \pm {0.95} \)	\( {92.98} \pm {0.31} \)
CUB	\( {96.14} \pm {0.35} \)	\( {89.66} \pm {1.53} \)	\( {92.28} \pm {0.74} \)	\( {86.37} \pm {1.18} \)	\( {92.19} \pm {0.77} \)	\( {74.24} \pm {5.68} \)	\( {99.17} \pm {0.14} \)
CUB-Incomp	\( {85.86} \pm {0.29} \)	\( {82.32} \pm {0.58} \)	\( {66.12} \pm {3.81} \)	\( {56.30} \pm {6.71} \)	\( {54.35} \pm {0.35} \)	\( {47.62} \pm {1.85} \)	\( {90.69} \pm {0.30} \)
CelebA	\( {52.01} \pm {0.45} \)	\( {38.82} \pm {1.32} \)	\( {27.19} \pm {1.17} \)	\( {21.20} \pm {2.74} \)	\( {27.79} \pm {1.07} \)	\( {29.11} \pm {0.40} \)	\( {62.01} \pm {1.21} \)
75%	MNIST-Add	\( {97.51} \pm {0.08} \)	\( {95.01} \pm {0.57} \)	\( {91.24} \pm {1.05} \)	\( {81.31} \pm {3.44} \)	\( {86.97} \pm {0.96} \)	\( {92.58} \pm {0.82} \)	\( {98.22} \pm {0.09} \)
MNIST-Add-Incomp	\( {93.48} \pm {0.25} \)	\( {90.35} \pm {0.96} \)	\( {88.09} \pm {1.03} \)	\( {78.62} \pm {1.90} \)	\( {88.15} \pm {0.68} \)	\( {90.09} \pm {0.91} \)	\( {94.18} \pm {0.24} \)
CUB	\( {98.98} \pm {0.14} \)	\( {93.95} \pm {1.95} \)	\( {95.73} \pm {1.00} \)	\( {91.18} \pm {1.61} \)	\( {96.97} \pm {0.53} \)	\( {79.81} \pm {4.96} \)	\( {99.82} \pm {0.08} \)
CUB-Incomp	\( {91.74} \pm {0.19} \)	\( {86.93} \pm {0.61} \)	\( {71.54} \pm {3.93} \)	\( {57.39} \pm {6.36} \)	\( {61.92} \pm {0.52} \)	\( {53.00} \pm {1.76} \)	\( {94.96} \pm {0.26} \)
CelebA	\( {56.99} \pm {0.37} \)	\( {41.59} \pm {1.39} \)	\( {28.04} \pm {1.42} \)	\( {23.70} \pm {3.82} \)	\( {28.63} \pm {1.48} \)	\( {30.04} \pm {0.48} \)	\( {66.48} \pm {1.10} \)
100%	MNIST-Add	\( {99.51} \pm {0.04} \)	\( {96.68} \pm {0.68} \)	\( {94.84} \pm {0.75} \)	\( {81.92} \pm {3.65} \)	\( {88.43} \pm {1.03} \)	\( {94.58} \pm {0.81} \)	\( {99.51} \pm {0.04} \)
MNIST-Add-Incomp	\( {94.99} \pm {0.11} \)	\( {91.53} \pm {1.01} \)	\( {91.28} \pm {0.81} \)	\( {79.21} \pm {1.98} \)	\( {89.36} \pm {0.69} \)	\( {91.54} \pm {1.03} \)	\( {94.99} \pm {0.11} \)
CUB	\( {99.90} \pm {0.04} \)	\( {96.75} \pm {1.72} \)	\( {96.42} \pm {1.24} \)	\( {95.03} \pm {1.95} \)	\( {98.97} \pm {0.35} \)	\( {82.81} \pm {3.72} \)	\( {99.90} \pm {0.04} \)
CUB-Incomp	\( {96.52} \pm {0.19} \)	\( {91.47} \pm {0.67} \)	\( {76.72} \pm {4.14} \)	\( {65.25} \pm {6.88} \)	\( {69.54} \pm {1.05} \)	\( {58.19} \pm {1.85} \)	\( {96.52} \pm {0.19} \)
CelebA	\( {70.02} \pm {0.62} \)	\( {48.10} \pm {1.72} \)	\( {29.38} \pm {1.80} \)	\( {30.75} \pm {9.89} \)	\( {29.43} \pm {2.02} \)	\( {31.04} \pm {0.22} \)	\( {70.02} \pm {0.62} \)