Original: D EBIASING C ONCEPT B OTTLENECK M ODELS WITH  I NSTRUMENTAL V ARIABLES
Revised: D EBIASING C ONCEPT B OTTLENECK M ODELS WITH A T ECHNIQUE FROM I NSTRUMENTAL V ARIABLES

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Concept-based explanation approach is a popular model interpertability tool because it expresses the reasons for a model’s predictions in terms of concepts that are meaningful for the domain experts.
Revised: Concept-based explanation approach is a popular model interpertability tool because it expresses the reasons for a model’s predictions in terms of concepts that are meaningful for the domain experts.

Original: In this work, we study the problem of the concepts being correlated with confounding information in the features.
Revised: In this work, we study the problem of the concepts being correlated with confounding information in the features.

Original: We propose a new causal prior graph for modeling the impacts of unobserved variables and a method to remove the impact of confounding information and noise using  the instrumental variable techniques.
Revised: We propose a new causal prior graph for modeling the impacts of unobserved variables and a method to remove the impact of confounding information and noise using a two -stage regression technique borrowed from the instrumental variable literature

Original: We also model the completeness of the concepts set and show that our debiasing method works when the concepts are not complete.
Revised: We also model the completeness of the concepts set and show that our debiasing method works when the concepts are not complete.

Original: Our synthetic and real-world experiments demonstrate the success of our method in removing biases and improving the ranking of the concepts in terms of their contribution to the explanation of the predictions.
Revised: Our synthetic and real-world experiments demonstrate the success of our method in removing biases and im - proving the ranking of the concepts in terms of their contribution to the explanation of the predictions.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Explaining the predictions of neural networks through higher level concepts (Kim et al., 2018; Ghorbani et al., 2019; Brocki & Chung, 2019; Hamidi-Haines et al., 2018) enables model interpretation on data with complex manifold structure such as images.
Revised: Explaining the predictions of neural networks through higher level concepts (Kim et al., 2018; Ghorbani et al., 2019; Brocki & Chung, 2019; Hamidi-Haines et al., 2018) enables model interpretation on data with complex manifold structure such as images.

Original: It also allows the use of domain knowledge during the explanation process.
Revised: It also allows the use of domain knowledge during the explanation process.

Original: The concept-based explanation has been used for medical imaging (Cai et al., 2019),
Revised: The concept-based explanation has been used for medical imaging (Cai et al., 2019),

Original: breast cancer histopathology (Graziani et al., 2018), cardiac MRIs (Clough et al., 2019), and meteorology (Sprague et al., 2019).
Revised: breast cancer histopathology (Graziani et al., 2018), cardiac MRIs (Clough et al., 2019), and meteorology (Sprague et al., 2019).

Original: When the set of concepts is carefully selected, we can estimate a model in which the discriminative information ﬂow from the feature vectors x through the concept vectors c and reach the labels y .
Revised: When the set of concepts is carefully selected, we can estimate a model in which the discriminative information ﬂow from the feature vectors x through the concept vectors c and reach the labels y .

Original: To this end, we train two models for prediction of the concept vectors from the features denoted by (cid:98) c ( x ) and the labels from the predicted concept vector (cid:98)
Revised: To this end, we train two models for prediction of the concept vectors from the features denoted by (cid:98) c ( x ) and the labels from the predicted concept vector (cid:98)

Original: y
Revised: y

Original: ( (cid:98) c ) .
Revised: ( (cid:98) c ) .

Original: This estimation process ensures that for each prediction we have the reasons for the prediction stated in terms of the predicted concept vector (cid:98) c ( x ) .
Revised: This estimation process ensures that for each prediction we have the reasons for the prediction stated in terms of the predicted concept vector (cid:98) c ( x ) .

Original: However, in reality, noise and confounding information (due to e.g. non-discriminative context) can inﬂuence both of the feature and concept vectors, resulting in confounded correlations between them.
Revised: However, in reality, noise and confounding information (due to e.g. non-discriminative context) can inﬂuence both of the feature and concept vectors, resulting in confounded correlations between them.

Original: Figure 1 provides an evidence for noise and confounding in the CUB-200-2011 dataset (Wah et al., 2011).
Revised: Figure 1 provides an evidence for noise and confounding in the CUB-200-2011 dataset (Wah et al., 2011).

Original: We train two predictors for the concepts vectors based on features (cid:98) c
Revised: We train two predictors for the concepts vectors based on features (cid:98) c

Original: ( x ) and labels (cid:98) c ( y ) and compare the Spearman correlation coefﬁcients between their predictions and the true ordinal value of the concepts.
Revised: ( x ) and labels (cid:98) c ( y ) and compare the Spearman correlation coefﬁcients between their predictions and the true ordinal value of the concepts.

Original: Having concepts for which (cid:98) c ( x ) is more accurate than (cid:98) c ( y ) could be due to noise, or due to hidden variables independent of the labels that spuriously correlated c and x , leading to undesirable explanations that include confounding or noise.
Revised: Having concepts for which (cid:98) c ( x ) is more accurate than (cid:98) c ( y ) could be due to noise, or due to hidden variables independent of the labels that spuriously correlated c and x , leading to undesirable explanations that include confounding or noise.

Original: In this work, using the Concept Bottleneck Models (CBM) (Koh et al., 2020; Losch et al., 2019) we demonstrate a method for removing the counfounding and noise (debiasing) the explanation with concept vectors and extend the results to Testing with Concept Activation Vectors (TCAV)
Revised: In this work, using the Concept Bottleneck Models (CBM) (Koh et al., 2020; Losch et al., 2019) we demonstrate a method for removing the counfounding and noise (debiasing) the explanation with concept vectors and extend the results to Testing with Concept Activation Vectors (TCAV)

Original: (Kim et al., 2018) technique.
Revised: (Kim et al., 2018) technique.

Original: We provide a new causal prior graph to account for the confounding information and concept completeness (Yeh et al., 2019).
Revised: We provide a new causal prior graph to account for the confounding information and concept completeness (Yeh et al., 2019).

Original: We describe the challenges in estimation of our causal prior graph and propose a new learning procedure.
Revised: We describe the challenges in estimation of our causal prior graph and propose a new learning procedure.

Original: Our estimation technique deﬁnes and predicts debiased concepts such that the predictive information of the features maximally ﬂow through them.
Revised: Our estimation technique deﬁnes and predicts debiased concepts such that the predictive information of the features maximally ﬂow through them.

Original: We show that using  thelabels as instrumental variables, we can successfully remove the impact of the confounding and noise from the predicted concept vectors.
Revised: We show that using a two-stage regression technique from the instrumental variables literature, we can successfully remove the impact of the confounding and noise from the predicted concept vectors.

Original: Our proposed procedure has three steps: (1) debias the concept vectors using the labels, (2) predictthe debiased concept vectors using the
Revised: Our proposed procedure has three steps: (1) debias the concept vectors using the labels, (2) predict

Original: 300 Concept ids 0 .0 .0 .0 .0 .0 .0 .
Revised: 300 Concept ids 0 .0 .0 .0 .0 .0 .0 .

Original: Sp e a r m a n C o rr
Revised: Sp e a r m a n C o rr

Revised: e l a t i o n ρ ( b c ( x ) , c ) ρ ( b c ( y ) , c ) c ( y ) for the 312 concepts in the test partition of the CUB-200-2011 dataset (Wah et al., 2011).

Revised: 112 concepts can be predicted more accurately with the features rather than the labels.

Revised: Concept ids in the x-axis are sorted in the increasing ρ ( (cid:98) c ( y ) , c ) order.

Revised: We provide the detailed steps to obtain the ﬁgure in Section 4.2.

Original: e l a t i o n ρ ( b c ( x ) , c ) ρ ( b c ( y ) , c ) features, and (3) use the predict concept vectors in the second step to predict the labels.
Revised: the debiased concept vectors using the features, and (3) use the predict concept vectors in the second step to predict the labels.

Original: Optionally, we can also ﬁnd the residual predictive information in the features that are not in the concepts.
Revised: Optionally, we can also ﬁnd the residual predictive information in the features that are not in the concepts.

Original: We validate the proposed method using a synthetic dataset and the CUB-200-2011 dataset.
Revised: We validate the proposed method using a synthetic dataset and the CUB-200-2011 dataset.

Original: On the synthetic data, we have access to the ground truth and show that in the presence of confounding and noise, our debiasing procedure improves the accuracy of recovering the true concepts.
Revised: On the synthetic data, we have access to the ground truth and show that in the presence of confounding and noise, our debiasing procedure improves the accuracy of recovering the true concepts.

Original: On the
Revised: On the

Original: CUB-200-2011 dataset, we use the RemOve And Retrain (ROAR) framework (Hooker et al., 2019) to show that our debiasing procedure ranks the concepts in the order of their explanation more accurately than the regular concept bottleneck models.
Revised: CUB-200-2011 dataset, we use the RemOve And Retrain (ROAR) framework (Hooker et al., 2019) to show that our debiasing procedure ranks the concepts in the order of their explanation more accurately than the regular concept bottleneck models.

Original: We also show that we improve the accuracy of CBNs in the prediction of labels using our debiasing technique.
Revised: We also show that we improve the accuracy of CBNs in the prediction of labels using our debiasing technique.

Original: Finally, using several examples, we also qualitatively show when the debasing helps improve the quality of concept-based explanations.
Revised: Finally, using several examples, we also qualitatively show when the debasing helps improve the quality of concept-based explanations.

Original:  M ETHODOLOGY
Revised:  M ETHODOLOGY

Original: Notations.
Revised: Notations.

Original: We follow the notation of Goodfellow et al.
Revised: We follow the notation of Goodfellow et al.

Original: and denote random vectors by bold font letters x and their values by bold symbols x .
Revised: and denote random vectors by bold font letters x and their values by bold symbols x .

Original: The notation p ( x ) is a probability measure on x and d p ( x = x ) is the inﬁnitesimal probability mass at x = x .
Revised: The notation p ( x ) is a probability measure on x and d p ( x = x ) is the inﬁnitesimal probability mass at x = x .

Original: We use (cid:98) y ( x ) to denote the the prediction of y given x .
Revised: We use (cid:98) y ( x ) to denote the the prediction of y given x .

Original: In the graphical models, we show the observed and unobserved variables using ﬁlled and hollow circles, respectively.
Revised: In the graphical models, we show the observed and unobserved variables using ﬁlled and hollow circles, respectively.

Original: Problem Statement.
Revised: Problem Statement.

Original: We assume that during the training phase, we are given triplets ( x i , c
Revised: We assume that during the training phase, we are given triplets ( x i , c

Original: i , y i ) for i = 1 , . . . , n data points.
Revised: i , y i ) for i = 1 , . . . , n data points.

Original: In addition to the regular features x and labels y , we are given a human interpretable concepts vector c for each data point.
Revised: In addition to the regular features x and labels y , we are given a human interpretable concepts vector c for each data point.

Original: Each element of the concept vector measures the degree of existence of the corresponding concept in the features.
Revised: Each element of the concept vector measures the degree of existence of the corresponding concept in the features.

Original: Thus, the concept vector typically have binary or ordinal values.
Revised: Thus, the concept vector typically have binary or ordinal values.

Original: Our goal is to learn to predict y as a function of x and use c for explaining the predictions.
Revised: Our goal is to learn to predict y as a function of x and use c for explaining the predictions.

Original: Performing in two steps, we ﬁrst learn a function (cid:98) c ( x ) and then learn another function (cid:98)
Revised: Performing in two steps, we ﬁrst learn a function (cid:98) c ( x ) and then learn another function (cid:98)

Original: y ( (cid:98) c
Revised: y ( (cid:98) c

Original: ( x )) .
Revised: ( x )) .

Original: The prediction (cid:98) c ( x ) is the explanation for our prediction (cid:98) y .
Revised: The prediction (cid:98) c ( x ) is the explanation for our prediction (cid:98) y .

Original: During the test time, only the features are given and the prediction+explanation algorithm predicts both (cid:98) y and (cid:98) c .
Revised: During the test time, only the features are given and the prediction+explanation algorithm predicts both (cid:98) y and (cid:98) c .

Original: In this paper, we aim to remove the bias and noise components from the estimated concept vector (cid:98) c such that it explains the reasons for prediction of the labels more accurately.
Revised: In this paper, we aim to remove the bias and noise components from the estimated concept vector (cid:98) c such that it explains the reasons for prediction of the labels more accurately.

Original: To this end, we ﬁrst need to propose a new causal prior graph that includes the potential unobserved confounders.
Revised: To this end, we ﬁrst need to propose a new causal prior graph that includes the potential unobserved confounders.

Original: 2. A N EW C AUSAL P RIOR G RAPH FOR CBM S
Revised: 2. A N EW C AUSAL P RIOR G RAPH FOR CBM S

Original: Figure 2a shows the ideal situation in explanation via high-level concepts.
Revised: Figure 2a shows the ideal situation in explanation via high-level concepts.

Original: The generative model corresponding to Figure 2a states that for generating each feature x
Revised: The generative model corresponding to Figure 2a states that for generating each feature x

Original: i
Revised: i

Original: we ﬁrst randomly draw the label y i .
Revised: we ﬁrst randomly draw the label y i .

Original: Given the label, we draw the concepts c i .
Revised: Given the label, we draw the concepts c i .

Original: Given the concepts, we generate the features.
Revised: Given the concepts, we generate the features.

Original: The
Revised: The

Original: Ideal Graphical Model
Revised: Ideal Graphical Model

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure]
Revised: [Figure]

Original: (cid:98) d ( y ) =
Revised: (cid:98) d ( y ) =

Original: E [ c | y ] inplace hierarchy in this graph is from nodes with less detailed information (labels) to more detailed ones (features, images).
Revised: E [ c | y ] in hierarchy in this graph is from nodes with less detailed information (labels) to more detailed ones (features, images).

Original: This model in Figure 2a is an explanation for the phenomenon in Figure 1, because the noise in generation of the concepts allows the x — c edge to be stronger than the c — y edge.
Revised: This model in Figure 2a is an explanation for the phenomenon in Figure 1, because the noise in generation of the concepts allows the x — c edge to be stronger than the c — y edge.

Original: However, another (non-mutually exclusive) explanation for this phenomenon is the existence of hidden confounders u shown in Figure 2b.
Revised: However, another (non-mutually exclusive) explanation for this phenomenon is the existence of hidden confounders u shown in Figure 2b.

Original: In this graphical model, u represents the confounders and d represents the unconfounded concepts.
Revised: In this graphical model, u represents the confounders and d represents the unconfounded concepts.

Original: Note that we assume that the confounders u and labels y are independent when x and c are not observed.
Revised: Note that we assume that the confounders u and labels y are independent when x and c are not observed.

Original: Another phenomenon captured in Figure 2b is the lack of concept completeness (Yeh et al., 2019).
Revised: Another phenomenon captured in Figure 2b is the lack of concept completeness (Yeh et al., 2019).

Original: It describes the situation when the features, compared to the concepts, have additional predictive information about the labels.
Revised: It describes the situation when the features, compared to the concepts, have additional predictive information about the labels.

Original: The non-linear structural equations corresponding to the causal prior graph in Figure 2b are as follows
Revised: The non-linear structural equations corresponding to the causal prior graph in Figure 2b are as follows

Original: [Equation]
Revised: [Equation]

Original: for some vector functions h , f 1 , f 2 , and f 3 .
Revised: for some vector functions h , f 1 , f 2 , and f 3 .

Original: We have ε d ⊥⊥ y and u ⊥⊥ y .
Revised: We have ε d ⊥⊥ y and u ⊥⊥ y .

Original: Our deﬁnition of d in Eq.
Revised: Our deﬁnition of d in Eq.

Original: (2) does not restrict u , because we simply attribute the difference between c and f 1 ( y ) to a function of the latent confounder u and noise.
Revised: (2) does not restrict u , because we simply attribute the difference between c and f 1 ( y ) to a function of the latent confounder u and noise.

Original: Our causal prior graph in Figure 2b corresponds to a generative process in which to generate an observed triplet ( x i , c
Revised: Our causal prior graph in Figure 2b corresponds to a generative process in which to generate an observed triplet ( x i , c

Original: i , y i )
Revised: i , y i )

Original: we ﬁrst draw a label y i and a confounder u i vector independently.
Revised: we ﬁrst draw a label y i and a confounder u i vector independently.

Original: Then we draw the discriminative concepts d i based on the label and generate the features x i jointly based on the concepts, label, and the confounder.
Revised: Then we draw the discriminative concepts d i based on the label and generate the features x i jointly based on the concepts, label, and the confounder.

Original: Finally, we draw the observed concept vector c i based on the drawn concept and confounder vectors.
Revised: Finally, we draw the observed concept vector c i based on the drawn concept and confounder vectors.

Original: Both causal graphs reﬂect our assumption that the direction of causality is from the labels to concepts and then to the features, y → d → x , to ensure that u and y are marginally independent in Figure 2b.
Revised: Both causal graphs reﬂect our assumption that the direction of causality is from the labels to concepts and then to the features, y → d → x , to ensure that u and y are marginally independent in Figure 2b.

Original: This direction also correspond to moving from more abstract class labels to concepts to detailed features.
Revised: This direction also correspond to moving from more abstract class labels to concepts to detailed features.

Original: During estimation, we ﬁt the functions in the x → d → y direction, because ﬁnding the statistical strength of an edge does not depend on its direction.
Revised: During estimation, we ﬁt the functions in the x → d → y direction, because ﬁnding the statistical strength of an edge does not depend on its direction.

Original: Estimation of the model in Figure 2b is challenging because there are two distinct paths for the information from the labels y to reach the features x .
Revised: Estimation of the model in Figure 2b is challenging because there are two distinct paths for the information from the labels y to reach the features x .

Original: Our solution is to prioritize the bottleneck path and estimate
Revised: Our solution is to prioritize the bottleneck path and estimate

Original: the y → d → x , then estimate the residuals of the regression using the y → x direct path.
Revised: the y → d → x , then estimate the residuals of the regression using the y → x direct path.

Original: Our two-stage estimation technique ensures that the predictive information of the features maximally ﬂow through the concepts.
Revised: Our two-stage estimation technique ensures that the predictive information of the features maximally ﬂow through the concepts.

Original: In the next sections, we focus on the ﬁrst stage and using  the instrumental variables  to eliminate the noise and confounding in estimation of the d → x link.
Revised: In the next sections, we focus on the ﬁrst phase and using a two-stage regression technique borrowed from the instrumental variables literature to eliminate the noise and confounding in estimation of the d → x link.

Original: 2. I NSTRUMENTAL V ARIABLES
Revised: 2. I NSTRUMENTAL V ARIABLES

Original: Background on Instrumental Variables.
Revised: Background on two-stage regression.

Original: In causal inference, instrumental variables (Stock, 2015;
Revised: In causal inference, instrumental variables (Stock, 2015;

Original: Pearl, 2009) denoted by z are commonly used to ﬁnd the causal impact of a variable x on y when x and y are jointly inﬂuenced by an unobserved confounder u (i.e., x ← u → y ).
Revised: Pearl, 2009) denoted by z are commonly used to ﬁnd the causal impact of a variable x on y when x and y are jointly inﬂuenced by an unobserved confounder u (i.e., x ← u → y ).

Original: The key requirement is that z should be correlated with x but independent of the confounding variable u (i.e. z → x → y and z ⊥⊥ u ).
Revised: The key requirement is that z should be correlated with x but independent of the confounding variable u (i.e. z → x → y and z ⊥⊥ u ).

Original: The commonly used 2-stage least squares ﬁrst regresses x in terms of z to obtain (cid:98) x followed by regression of y in terms of (cid:98) x .
Revised: The commonly used 2-stage least squares ﬁrst regresses x in terms of z to obtain (cid:98) x followed by regression of y in terms of (cid:98) x .

Original: Because of independence between z and u , (cid:98) x is also independent of u .
Revised: Because of independence between z and u , (cid:98) x is also independent of u .

Original: Thus, in the second regression the confounding impact of u is eliminated.
Revised: Thus, in the second regression the confounding impact of u is eliminated.

Original: Our goal is to use the instrumental variable trick again to remove the confounding factors impacting features and concept vectors.
Revised: Our goal is to use the two-stage regression trick again to remove the confounding factors impacting features and concept vectors.

Original: Instrumental Variables for CBMs.
Revised: Two-Stage Regression for CBMs.

Original: In our causal graph in Figure 2b, the label y is a valid instrument for the study of the relationship between concepts d and features x .
Revised: In our causal graph in Figure 2b, the label y can be used for the study of the relationship between concepts d and features x .

Original: We predict d as a function of y and use it in place of the concepts in the concept bottleneck models.
Revised: We predict d as a function of y and use it in place of the concepts in the concept bottleneck models.

Original: The graphical model corresponding to this procedure is shown in Figure 2c, where the link u → c is eliminated.
Revised: The graphical model corresponding to this procedure is shown in Figure 2c, where the link u → c is eliminated.

Original: In particular, given the independence relationship y ⊥⊥ u , we have (cid:98) d ( y ) =
Revised: In particular, given the independence relationship y ⊥⊥ u , we have (cid:98) d ( y ) =

Original: E [ c | y ] ⊥⊥ h ( u ) .
Revised: E [ c | y ] ⊥⊥ h ( u ) .

Original: This is the basis for our debiasing method in the next section.
Revised: This is the basis for our debiasing method in the next section.

Original: 2. T HE E STIMATION M ETHOD
Revised: 2. T HE E STIMATION M ETHOD

Original: Our estimation uses the observation that in graph 2b the label vector
Revised: Our estimation uses the observation that in graph 2b the label vector

Original: y is a valid instrument for removing the correlations due to u .
Revised: y is a valid instrument for removing the correlations due to u .

Original: Combining Eqs.
Revised: Combining Eqs.

Original: (1) and (2) we have c = f 1 ( y )
Revised: (1) and (2) we have c = f 1 ( y )

Original: + h ( u ) + ε d .
Revised: + h ( u ) + ε d .

Original: Taking expectation with respect to p ( c | y ) , we have
Revised: Taking expectation with respect to p ( c | y ) , we have

Original: [Equation]
Revised: [Equation]

Original: The last step is because both u and ε d are independent of y .
Revised: The last step is because both u and ε d are independent of y .

Original: Thus, two terms are constant in terms of x and y and can be eliminated after estimation.
Revised: Thus, two terms are constant in terms of x and y and can be eliminated after estimation.

Original: Eq.
Revised: Eq.

Original: (4) allows us to remove the impact of u and ε d and estimate the denoised and debiased (cid:98) d ( y )
Revised: (4) allows us to remove the impact of u and ε d and estimate the denoised and debiased (cid:98) d ( y )

Original: =
Revised: =

Original: E [ c | y ] .
Revised: E [ c | y ] .

Original: We ﬁnd E
Revised: We ﬁnd E

Original: [ c | y ] using a neural network trained on ( c i , y i ) pairs and use them as pseudo-observations in place of d i .
Revised: [ c | y ] using a neural network trained on ( c i , y i ) pairs and use them as pseudo-observations in place of d i .

Original: Given our debiased prediction for the discriminative concepts d i , we can perform the CBMs’ two-steps of x → d and d → y estimation.
Revised: Given our debiased prediction for the discriminative concepts d i , we can perform the CBMs’ two-steps of x → d and d → y estimation.

Original: Because we use expected values of c in place of d during the learning process (i.e., (cid:98) d ( y )
Revised: Because we use expected values of c in place of d during the learning process (i.e., (cid:98) d ( y )

Original: =
Revised: =

Original: E [ c | y ] ), the debiased concept vectors have values within the ranges of original concept vectors c .
Revised: E [ c | y ] ), the debiased concept vectors have values within the ranges of original concept vectors c .

Original: Thus, we do not lose the human readability with the debiased concept vectors.
Revised: Thus, we do not lose the human readability with the debiased concept vectors.

Original: Incorporating Uncertainty in Prediction of Concepts.
Revised: Incorporating Uncertainty in Prediction of Concepts.

Original: Our empirical observations show that prediction of the concepts from the features can be highly uncertain.
Revised: Our empirical observations show that prediction of the concepts from the features can be highly uncertain.

Original: Hence, we present a CBM estimator that takes into account the uncertainties in prediction of the concepts.
Revised: Hence, we present a CBM estimator that takes into account the uncertainties in prediction of the concepts.

Original: We take the conditional expectation of the labels y given features x as follows
Revised: We take the conditional expectation of the labels y given features x as follows

Original: [Equation]
Revised: [Equation]

Original: where p φ ( d | x ) is the probability function, parameterized by φ , that captures the uncertainty in prediction of labels from features.
Revised: where p φ ( d | x ) is the probability function, parameterized by φ , that captures the uncertainty in prediction of labels from features.

Original: The g θ ( · ) function predicts labels from the debiased concepts.
Revised: The g θ ( · ) function predicts labels from the debiased concepts.

Original: In summary, we perform the steps in Algorithm 1 to learn debiased CBMs.
Revised: In summary, we perform the steps in Algorithm 1 to learn debiased CBMs.

Original: In Line 3, we approximate the integral using Monte Carlo approach by drawing from the distribution p (cid:98) φ ( d | x ) estimated in Line
Revised: In Line 3, we approximate the integral using Monte Carlo approach by drawing from the distribution p (cid:98) φ ( d | x ) estimated in Line

Original: 2, see (Hartford et al., 2017).
Revised: 2, see (Hartford et al., 2017).

Original: Given the labels’ data type, we can use the appropriate loss functions and are not limited to the least squares loss.
Revised: Given the labels’ data type, we can use the appropriate loss functions and are not limited to the least squares loss.

Original: Line 4 removes the constant mean that can be due to noise or confounding, as we discussed after Eq.
Revised: Line 4 removes the constant mean that can be due to noise or confounding, as we discussed after Eq.

Original: In the common case of binary concept vectors, we use the debiased concepts estimated in Line 4 to rank the concepts in the order of their contribution to explanation of the predictions.
Revised: In the common case of binary concept vectors, we use the debiased concepts estimated in Line 4 to rank the concepts in the order of their contribution to explanation of the predictions.

Original:  D ISCUSSIONS
Revised:  D ISCUSSIONS

Original: Debiasing TCAVs.
Revised: Debiasing TCAVs.

Original: While we presented our debiasing algorithm for the CBMs, we can easily use it to debias the TCAV
Revised: While we presented our debiasing algorithm for the CBMs, we can easily use it to debias the TCAV

Original: (Kim et al., 2018) explanations too.
Revised: (Kim et al., 2018) explanations too.

Original: We can use the ﬁrst step to remove the bias due to the confounding and perform TCAV using (cid:98) d vectors, instead of c vectors.
Revised: We can use the ﬁrst step to remove the bias due to the confounding and perform TCAV using (cid:98) d vectors, instead of c vectors.

Original: The TCAV method is attractive, because unlike CBMs, it analyzes the existing neural networks and does not need to deﬁne a new model.
Revised: The TCAV method is attractive, because unlike CBMs, it analyzes the existing neural networks and does not need to deﬁne a new model.

Original: Measuring Concept Completeness.
Revised: Measuring Concept Completeness.

Original: Our primary objective of modeling the concept incompleteness phenomenon in our causal prior graph is to show that our debiasing method does not need the concept completeness assumption to work properly.
Revised: Our primary objective of modeling the concept incompleteness phenomenon in our causal prior graph is to show that our debiasing method does not need the concept completeness assumption to work properly.

Original: If we are interested in measuring the degree of completeness of the concepts, we can do it based on the deﬁnition by Yeh et al.
Revised: If we are interested in measuring the degree of completeness of the concepts, we can do it based on the deﬁnition by Yeh et al.

Original: To this end, we ﬁt a unrestricted neural network q ( x ) =
Revised: To this end, we ﬁt a unrestricted neural network q ( x ) =

Original: E [ y | x ] to the to the residuals in the step 3 of our debiasing algorithm.
Revised: E [ y | x ] to the to the residuals in the step 3 of our debiasing algorithm.

Original: The function q ( · ) captures the residual information in x .
Revised: The function q ( · ) captures the residual information in x .

Original: We compare the improvement in prediction accuracy over the accuracy in step 3 to quantify the degree of concept incompleteness.
Revised: We compare the improvement in prediction accuracy over the accuracy in step 3 to quantify the degree of concept incompleteness.

Original: Because we ﬁrst predict the labels y using the CBM and then ﬁt the q ( x ) to the residuals, we ensure that the predictive information maximally go through the CBM link x ← d ← y .
Revised: Because we ﬁrst predict the labels y using the CBM and then ﬁt the q ( x ) to the residuals, we ensure that the predictive information maximally go through the CBM link x ← d ← y .

Original: Prior Work on Causal Concept-Based Explanation.
Revised: Prior Work on Causal Concept-Based Explanation.

Original: Among the existing works on causal concept-based explanation, Goyal et al.
Revised: Among the existing works on causal concept-based explanation, Goyal et al.

Original: (2019) propose a different causal prior graph to model the spurious correlations among the concepts and remove them using conditional variational autoencoders.
Revised: (2019) propose a different causal prior graph to model the spurious correlations among the concepts and remove them using conditional variational autoencoders.

Original: In contrast, we aim at handling noise and spurious correlations between the features and concepts using the labels as instruments.
Revised: In contrast, we aim at handling noise and spurious correlations between the features and concepts using the labels as instruments.

Original: Which work is more appropriate for a problem depending on the assumptions underlying that problem.
Revised: Which work is more appropriate for a problem depending on the assumptions underlying that problem.

Original: The Special Linear Gaussian Case.
Revised: The Special Linear Gaussian Case.

Original: When the concepts have real continuous values, we might use a simple multivariate Gaussian distribution to model p ( d | x ) =
Revised: When the concepts have real continuous values, we might use a simple multivariate Gaussian distribution to model p ( d | x ) =

Original: N ( x , σ I ) , σ > 0 .
Revised: N ( x , σ I ) , σ > 0 .

Original: If we further use a linear regression to predict labels from the concepts, we can show that the steps are simpliﬁed as follows:
Revised: If we further use a linear regression to predict labels from the concepts, we can show that the steps are simpliﬁed as follows:

Original: Learn (cid:98) d
Revised: Learn (cid:98) d

Original: ( y ) by predicting y i
Revised: ( y ) by predicting y i

Original: → c i . 2.
Revised: → c i . 2.

Original: Learn ˆˆ d ( x ) by predicting x i → (cid:98) d i . 3.
Revised: Learn ˆˆ d ( x ) by predicting x i → (cid:98) d i . 3.

Original: Learn (cid:98) y
Revised: Learn (cid:98) y

Original: ( ˆ ˆ d ) by predicting ˆ ˆ d i → y i .
Revised: ( ˆ ˆ d ) by predicting ˆ ˆ d i → y i .

Original: The above steps show that given the linear Gaussian assumptions, steps 2 and 3 coincide with the independent bottleneck (Koh et al., 2020) training of CBMs.
Revised: The above steps show that given the linear Gaussian assumptions, steps 2 and 3 coincide with the independent bottleneck (Koh et al., 2020) training of CBMs.

Original: We only need to change the concepts from c i to (cid:98)
Revised: We only need to change the concepts from c i to (cid:98)

Original: d i we can eliminate the noise and confounding bias.
Revised: d i we can eliminate the noise and confounding bias.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: Evaluation of explanation algorithms is notoriously difﬁcult.
Revised: Evaluation of explanation algorithms is notoriously difﬁcult.

Original: Thus, we ﬁrst present experiments with synthetic data to show that our debiasing technique improves the explanation accuracy when we know the true explanations.
Revised: Thus, we ﬁrst present experiments with synthetic data to show that our debiasing technique improves the explanation accuracy when we know the true explanations.

Original: Then, on the CUB-200-2011 dataset, we use the ROAR (Hooker et al., 2019) framework to show that the debiasing improves the explanation accuracy.
Revised: Then, on the CUB-200-2011 dataset, we use the ROAR (Hooker et al., 2019) framework to show that the debiasing improves the explanation accuracy.

Original: Finally, using several examples, we identify the circumstances in which the debiasing helps.
Revised: Finally, using several examples, we identify the circumstances in which the debiasing helps.

Original: 4. S YNTHETIC D ATA E XPERIMENTS
Revised: 4. S YNTHETIC D ATA E XPERIMENTS

Original: We create a synthetic dataset according to the following steps:
Revised: We create a synthetic dataset according to the following steps:

Original: [Figure]
Revised: [Figure]

Original: Generate n vectors
Revised: Generate n vectors

Original: y i ∈
Revised: y i ∈

Original: R 100 with elements distributed according to unit normal distribution N (0 , 1) .
Revised: R 100 with elements distributed according to unit normal distribution N (0 , 1) .

Original: Generate n vectors u i ∈ R 100 with elements distributed according to unit normal distribution N (0 , 1) .
Revised: Generate n vectors u i ∈ R 100 with elements distributed according to unit normal distribution N (0 , 1) .

Original: Generate n vectors ε c,i ∈ R 100 with elements distributed according to scaled normal distribution N (0 , σ = 0 . 02) . Generate n vectors ε x,i ∈ R 100 with elements distributed according to scaled normal distribution N (0 , σ = 0 . 02) .
Revised: Generate n vectors ε c,i ∈ R 100 with elements distributed according to scaled normal distribution N (0 , σ = 0 . 02) . Generate n vectors ε x,i ∈ R 100 with elements distributed according to scaled normal distribution N (0 , σ = 0 . 02) .

Original: Generate matrices W 1 , W 2 , W 3 , W 4 ∈ R 100 × 100 with elements distributed according to scaled normal distribution N (0 , σ = 0 . 1) .
Revised: Generate matrices W 1 , W 2 , W 3 , W 4 ∈ R 100 × 100 with elements distributed according to scaled normal distribution N (0 , σ = 0 . 1) .

Original: Compute d i =
Revised: Compute d i =

Original: W 1 y i + ε d,i for i = 1 , . . . , n . 7.
Revised: W 1 y i + ε d,i for i = 1 , . . . , n . 7.

Original: Compute c i =
Revised: Compute c i =

Original: d i + W 2 u i for i = 1 , . . . , n . 8.
Revised: d i + W 2 u i for i = 1 , . . . , n . 8.

Original: Compute x i =
Revised: Compute x i =

Original: W 3 d
Revised: W 3 d

Original: i + W 4 u i + ε x,i for i = 1 , . . . , n .
Revised: i + W 4 u i + ε x,i for i = 1 , . . . , n .

Original: In Figure 3a, we plot the correlation between the true unconfounded and noiseless concepts W y and the estimated concept vectors with the regular two-step procedure (without debiasing) and our debiasing method, as a function of sample size n .
Revised: In Figure 3a, we plot the correlation between the true unconfounded and noiseless concepts W y and the estimated concept vectors with the regular two-step procedure (without debiasing) and our debiasing method, as a function of sample size n .

Original: The results show that the bias due to confounding does not vanish as we increase the sample size and our debiasing technique can make the results closer to the true discriminative concepts.
Revised: The results show that the bias due to confounding does not vanish as we increase the sample size and our debiasing technique can make the results closer to the true discriminative concepts.

Original: In the ideal case, when the confounding impact is orthogonal to the impact of the labels, our debiasing algorithm recovers the true concepts more accurately.
Revised: In the ideal case, when the confounding impact is orthogonal to the impact of the labels, our debiasing algorithm recovers the true concepts more accurately.

Original: Figure 3b demonstrates the scenario when
Revised: Figure 3b demonstrates the scenario when

Original: W 1 ⊥ W 2 , W 4 in the synthetic data generation.
Revised: W 1 ⊥ W 2 , W 4 in the synthetic data generation.

Original: To generate matrices with this constraint, we ﬁrst generate a unitary matrix Q and generate four diagonal matrices Λ 1 , . . .
Revised: To generate matrices with this constraint, we ﬁrst generate a unitary matrix Q and generate four diagonal matrices Λ 1 , . . .

Original: , Λ 4 with diagonal elements drawn from Uniform(0 . 1 , 1) .
Revised: , Λ 4 with diagonal elements drawn from Uniform(0 . 1 , 1) .

Original: This choice of the distribution for the diagonal elements caps the condition number of the matrices by 10 .
Revised: This choice of the distribution for the diagonal elements caps the condition number of the matrices by 10 .

Original: To satisfy the orthogonality constraint, we set the ﬁrst 50 diagonal elements of Λ 2 and Λ 4 and last 50 diagonal elements of Λ 1 to zero.
Revised: To satisfy the orthogonality constraint, we set the ﬁrst 50 diagonal elements of Λ 2 and Λ 4 and last 50 diagonal elements of Λ 1 to zero.

Original: We compute the matrices as W j = Q Λ j Q (cid:62) for j = 1 , . . .
Revised: We compute the matrices as W j = Q Λ j Q (cid:62) for j = 1 , . . .

Original: The orthogonality allows perfect separation of u and y impacts and the perfect debiasing by our method as the sample size grows.
Revised: The orthogonality allows perfect separation of u and y impacts and the perfect debiasing by our method as the sample size grows.

Original: In Figure 6, we repeat the synthetic experiments in a higher noise regime where the standard deviation of the noise variables ε c and ε c is set to σ = 0 .
Revised: In Figure 6, we repeat the synthetic experiments in a higher noise regime where the standard deviation of the noise variables ε c and ε c is set to σ = 0 .

Original: The results conﬁrm our main claim about our debiasing algorithm.
Revised: The results conﬁrm our main claim about our debiasing algorithm.

Original: 4. CUB-200-2011 E XPERIMENTS
Revised: 4. CUB-200-2011 E XPERIMENTS

Original: Dataset and preprocessing.
Revised: Dataset and preprocessing.

Original: We evaluate the performance of the proposed approach on the CUB200-2011 dataset (Wah et al., 2011).
Revised: We evaluate the performance of the proposed approach on the CUB200-2011 dataset (Wah et al., 2011).

Original: The dataset includes 11788 pictures (in 5994/5794 train/test
Revised: The dataset includes 11788 pictures (in 5994/5794 train/test

Original: [Table]
Revised: [Table]

Original: partitions) of 200 different types of birds, annotated both for the bird type and 312 different concepts about each picture.
Revised: partitions) of 200 different types of birds, annotated both for the bird type and 312 different concepts about each picture.

Original: The concept annotations are binary, whether the concept exists or not.
Revised: The concept annotations are binary, whether the concept exists or not.

Original: However, for each statement, a four-level certainty score has been also assigned: 1: not visible, 2: guessing, 3: probably, and 4: deﬁnitely.
Revised: However, for each statement, a four-level certainty score has been also assigned: 1: not visible, 2: guessing, 3: probably, and 4: deﬁnitely.

Original: We combine the binary annotation and the certainty score to create a 7-level ordinal variable as the annotation for each image as summarized in Table 1.
Revised: We combine the binary annotation and the certainty score to create a 7-level ordinal variable as the annotation for each image as summarized in Table 1.

Original: For simplicity, we map the 7-level ordinal values to uniformly spaced valued in the [0 , 1] interval.
Revised: For simplicity, we map the 7-level ordinal values to uniformly spaced valued in the [0 , 1] interval.

Original: We randomly choose 15% of the training set and hold out as the validation set.
Revised: We randomly choose 15% of the training set and hold out as the validation set.

Original: The result in Figure 1.
Revised: The result in Figure 1.

Original: To compare the association strength between y and c with the association strength between x and c we train two predictors of concepts (cid:98) c
Revised: To compare the association strength between y and c with the association strength between x and c we train two predictors of concepts (cid:98) c

Original: ( x ) and (cid:98) c
Revised: ( x ) and (cid:98) c

Original: ( y ) .
Revised: ( y ) .

Original: We use TorchVision’s pre-trained ResNet152 network (He et al., 2016) for prediction of the concepts from the images.
Revised: We use TorchVision’s pre-trained ResNet152 network (He et al., 2016) for prediction of the concepts from the images.

Original: Because the labels y have categorical values, (cid:98) c
Revised: Because the labels y have categorical values, (cid:98) c

Original: ( y ) is simply the average concept annotation scores per each class.
Revised: ( y ) is simply the average concept annotation scores per each class.

Original: We use the Spearman correlation to ﬁnd the association strengths in pairs ( (cid:98) c ( x ) , c ) and ( (cid:98) c
Revised: We use the Spearman correlation to ﬁnd the association strengths in pairs ( (cid:98) c ( x ) , c ) and ( (cid:98) c

Original: ( y ) , c ) because the concept annotations are ordinal numbers.
Revised: ( y ) , c ) because the concept annotations are ordinal numbers.

Original: The concept ids in the x-axis are sorted in terms of increasing values of ρ ( (cid:98) c ( y ) , c ) .
Revised: The concept ids in the x-axis are sorted in terms of increasing values of ρ ( (cid:98) c ( y ) , c ) .

Original: The top ten concepts with the largest values of ρ ( (cid:98) c ( x ) , c ) − ρ ( (cid:98) c ( y ) , c ) are ‘has back color::green’, ‘has upper tail color::green’, ‘has upper tail color::orange’, ‘has upper tail color::pink’, ‘has back color::rufous’, ‘has upper tail color::purple’, ‘has back color::pink’, ‘has upper tail color::iridescent’, ‘has back color::purple’, and ‘has back color::iridescent’.
Revised: The top ten concepts with the largest values of ρ ( (cid:98) c ( x ) , c ) − ρ ( (cid:98) c ( y ) , c ) are ‘has back color::green’, ‘has upper tail color::green’, ‘has upper tail color::orange’, ‘has upper tail color::pink’, ‘has back color::rufous’, ‘has upper tail color::purple’, ‘has back color::pink’, ‘has upper tail color::iridescent’, ‘has back color::purple’, and ‘has back color::iridescent’.

Original: These concepts are all related to color and can be easily confounded by the context of the images.
Revised: These concepts are all related to color and can be easily confounded by the context of the images.

Original: Training details for Algorithm 1.
Revised: Training details for Algorithm 1.

Original: We model the distribution of the concept logits as Gaussians with means equal to the ResNet152’s logit outputs and a diagonal covariance matrix.
Revised: We model the distribution of the concept logits as Gaussians with means equal to the ResNet152’s logit outputs and a diagonal covariance matrix.

Original: We estimate the variance for each concept by using the logits of the true concept annotation scores that are clamped into [0 . 05 , 0 .
Revised: We estimate the variance for each concept by using the logits of the true concept annotation scores that are clamped into [0 . 05 , 0 .

Original: 95] to avoid large logit numbers.
Revised: 95] to avoid large logit numbers.

Original: In each iteration of the training loop for Line 3, we draw 25 samples from the estimated p ( d | x ) .
Revised: In each iteration of the training loop for Line 3, we draw 25 samples from the estimated p ( d | x ) .

Original: Predictor of labels from concepts (the function g ( · ) in
Revised: Predictor of labels from concepts (the function g ( · ) in

Original: Eq.
Revised: Eq.

Original: (5)) is a three-layer feed-forward neural network with hidden layer sizes (312, 312, 200).
Revised: (5)) is a three-layer feed-forward neural network with hidden layer sizes (312, 312, 200).

Original: There is a skip connection from the input to the penultimate layer.
Revised: There is a skip connection from the input to the penultimate layer.

Original: All algorithms are trained with Adam optimization algorithm (Kingma & Ba, 2014).
Revised: All algorithms are trained with Adam optimization algorithm (Kingma & Ba, 2014).

Original: Quantitative Results.
Revised: Quantitative Results.

Original: Comparing to the baseline algorithm, our debiasing technique increases the average Spearman correlation between (cid:98) c ( x ) and (cid:98) c
Revised: Comparing to the baseline algorithm, our debiasing technique increases the average Spearman correlation between (cid:98) c ( x ) and (cid:98) c

Original: ( y ) from 0.406 to 0.508.
Revised: ( y ) from 0.406 to 0.508.

Original: For the above 10 concepts, our algorithm increases the average Spearman correlation from 0.283 to 0.389.
Revised: For the above 10 concepts, our algorithm increases the average Spearman correlation from 0.283 to 0.389.

Original: Our debiasing algorithm also improves the generalization in prediction of the image labels.
Revised: Our debiasing algorithm also improves the generalization in prediction of the image labels.

Original: The debiasing also improves the top-5 accuracy of predicting the labels from 39.5% to 49.3%.
Revised: The debiasing also improves the top-5 accuracy of predicting the labels from 39.5% to 49.3%.

Original: To show that our proposed debiasing accurately ranks the concepts in terms of their explanation of the predictions, we use the RemOve And Retrain (ROAR) framework (Hooker et al., 2019).
Revised: To show that our proposed debiasing accurately ranks the concepts in terms of their explanation of the predictions, we use the RemOve And Retrain (ROAR) framework (Hooker et al., 2019).

Original: In the
Revised: In the

Original: ROAR framework, we sort the concept using the scores E [ d | x i ] − 1 n (cid:80) ni
Revised: ROAR framework, we sort the concept using the scores E [ d | x i ] − 1 n (cid:80) ni

Original: =1 E [ d | x i ] in the ascending order.
Revised: =1 E [ d | x i ] in the ascending order.

Original: Then, we mask (set to zero) the least explanatory x % of the concepts using the scores and retrain the g θ ( d ) function.
Revised: Then, we mask (set to zero) the least explanatory x % of the concepts using the scores and retrain the g θ ( d ) function.

Original: We perform the procedure for x ∈ { 0 , 10 , 20 , . .
Revised: We perform the procedure for x ∈ { 0 , 10 , 20 , . .

Original: , 80 , 90 , 95 } and record the testing top-5 accuracy of predicting the labels y .
Revised: , 80 , 90 , 95 } and record the testing top-5 accuracy of predicting the labels y .

Original: We repeat the ROAR experiments for 3 times and report the average accuracy as we vary the masking percentage.
Revised: We repeat the ROAR experiments for 3 times and report the average accuracy as we vary the masking percentage.

Original: [Figure]
Revised: [Figure]

Original: Figure 4 shows the ROAR evaluation of the regular and debiased algorithms.
Revised: Figure 4 shows the ROAR evaluation of the regular and debiased algorithms.

Original: Because the debiased algorithm is more accurate, for easier comparison, we normalize both curves by dividing them by their accuracy at masking percentage x = 0% .
Revised: Because the debiased algorithm is more accurate, for easier comparison, we normalize both curves by dividing them by their accuracy at masking percentage x = 0% .

Original: An immediate observation is that the plot for debiased algorithm stays above the regular one, which is a clear indication of its superior performance in identifying the least explanatory concepts.
Revised: An immediate observation is that the plot for debiased algorithm stays above the regular one, which is a clear indication of its superior performance in identifying the least explanatory concepts.

Original: The results show several additional interesting insights too.
Revised: The results show several additional interesting insights too.

Original: First, the prediction of the bird species largely rely on a sparse set of concepts as we can mask 95% of the concepts and still have a decent accuracy.
Revised: First, the prediction of the bird species largely rely on a sparse set of concepts as we can mask 95% of the concepts and still have a decent accuracy.

Original: Second, masking a small percentage of irrelevant concepts reduces the noise in the features and improves the generalization performance of both algorithms.
Revised: Second, masking a small percentage of irrelevant concepts reduces the noise in the features and improves the generalization performance of both algorithms.

Original: Our debiased algorithm is more successful by being faster at ﬁnding the noisy features before x = 20% masking.
Revised: Our debiased algorithm is more successful by being faster at ﬁnding the noisy features before x = 20% masking.

Original: Finally, the debiased accuracy curve is less steep after x = 80% , which again indicates its success in ﬁnding the most explanatory concepts.
Revised: Finally, the debiased accuracy curve is less steep after x = 80% , which again indicates its success in ﬁnding the most explanatory concepts.

Original: Qualitative analysis of the results.
Revised: Qualitative analysis of the results.

Original: In Figure 5, we show 12 images for which the (cid:98) d and c are signiﬁcantly different.
Revised: In Figure 5, we show 12 images for which the (cid:98) d and c are signiﬁcantly different.

Original: A common pattern among the examples is that the context of the image does not allow accurate annotations by the annotators.
Revised: A common pattern among the examples is that the context of the image does not allow accurate annotations by the annotators.

Original: In images 3, 4, 5, 6, 7, 11, and 12 in Figure 5, the ten color-related concepts listed at the beginning are all set to 0.5, indicating that the annotators have failed in annotation.
Revised: In images 3, 4, 5, 6, 7, 11, and 12 in Figure 5, the ten color-related concepts listed at the beginning are all set to 0.5, indicating that the annotators have failed in annotation.

Original: However, our algorithm correctly identiﬁes that for example Ivory Gulls do not have green-colored backs by predicting (cid:98) c
Revised: However, our algorithm correctly identiﬁes that for example Ivory Gulls do not have green-colored backs by predicting (cid:98) c

Original: = 0 .
Revised: = 0 .

Original: 08 which is closer to (cid:98)
Revised: 08 which is closer to (cid:98)

Original: c
Revised: c

Original: ( y )
Revised: ( y )

Original: = 0 .
Revised: = 0 .

Original: 06 than the true c = 0 .
Revised: 06 than the true c = 0 .

Original: Another pattern is the impact of the color of the environment on the accuracy of the annotations.
Revised: Another pattern is the impact of the color of the environment on the accuracy of the annotations.

Original: For example, the second image from the left is an image of Pelagic cormorant, whose back and upper tail colors are unlikely to be green with per-class average of 0 .
Revised: For example, the second image from the left is an image of Pelagic cormorant, whose back and upper tail colors are unlikely to be green with per-class average of 0 .

Original: 12 and 0 .
Revised: 12 and 0 .

Original: 07 , respectively.
Revised: 07 , respectively.

Original: However, because of the color of the image and the reﬂections, the annotator has assigned 1 . 0 to both of ‘has back color::green’ and ‘has upper tail color::green’ concepts.
Revised: However, because of the color of the image and the reﬂections, the annotator has assigned 1 . 0 to both of ‘has back color::green’ and ‘has upper tail color::green’ concepts.

Original: Our algorithm predicts 0 .
Revised: Our algorithm predicts 0 .

Original: 11 and 0 .for
Revised: 11 and 0 .for

Original: these two features respectively, which are closer to the per-class average.
Revised: these two features respectively, which are closer to the per-class average.

Original: [Table]
Revised: [Table]

Original: In Table 2, we list six examples to show the superior accuracy of the debiased CBM in ranking the concepts in terms of their explanation power.
Revised: In Table 2, we list six examples to show the superior accuracy of the debiased CBM in ranking the concepts in terms of their explanation power.

Original: Moreover, in Table 3 in the appendix, we list the top and bottom concepts in terms of their predictive uncertainty in our debiased CBM.
Revised: Moreover, in Table 3 in the appendix, we list the top and bottom concepts in terms of their predictive uncertainty in our debiased CBM.

Original:  C ONCLUSIONS AND F UTURE W ORKS
Revised:  C ONCLUSIONS AND F UTURE W ORKS

Original: Studying the concept-based explanation techniques, we provided evidences for potential existence of spurious association between the features and concepts due to unobserved latent variables or noise.
Revised: Studying the concept-based explanation techniques, we provided evidences for potential existence of spurious association between the features and concepts due to unobserved latent variables or noise.

Original: We proposed a new causal prior graph that models the impact of the noise and latent confounding fron the estimated concepts.
Revised: We proposed a new causal prior graph that models the impact of the noise and latent confounding fron the estimated concepts.

Original: We showed that using the labels as instruments, we can remove the impact of the context from the explanations.
Revised: We showed that using the labels as instruments, we can remove the impact of the context from the explanations.

Original: Our experiments showed that our debiasing technique not only improves the quality of the explanations, but also improve the accuracy of predicting labels through the concepts.
Revised: Our experiments showed that our debiasing technique not only improves the quality of the explanations, but also improve the accuracy of predicting labels through the concepts.

Original: As future work, we will investigate other instrumental variable techniques to ﬁnd the most accurate debiasing method.
Revised: As future work, we will investigate other two-stage-regression techniques to ﬁnd the most accurate debiasing method.

Original: A H IGH - NOISE S YNTHETIC EXPERIMENTS
Revised: A H IGH - NOISE S YNTHETIC EXPERIMENTS

Original: [Figure]
Revised: [Figure]

Original: Certainty Level Concepts
Revised: Certainty Level Concepts

Original: Most Uncertain (194) has nape color::black, (260) has primary color::black, (164) has forehead color::black, (7) has bill shape::all-purpose, (132) has throat color::black, (305) has crown color::black, (133) has throat color::white, (150) has bill length::about the same as head, (8) has bill shape::cone, (152) has bill length::shorter than head Least Uncertain (216) has wing shape::tapered-wings, (215) has wing shape::broad-wings, (217) has wing shape::long-wings, (214) has wing shape::pointed-wings, (77) has tail shape::fan-shaped tail, (213) has wing shape::rounded-wings, (79) has tail shape::squared tail, (83) has upper tail color::purple, (82) has upper tail color::iridescent, (75) has tail shape::rounded tail
Revised: Most Uncertain (194) has nape color::black, (260) has primary color::black, (164) has forehead color::black, (7) has bill shape::all-purpose, (132) has throat color::black, (305) has crown color::black, (133) has throat color::white, (150) has bill length::about the same as head, (8) has bill shape::cone, (152) has bill length::shorter than head Least Uncertain (216) has wing shape::tapered-wings, (215) has wing shape::broad-wings, (217) has wing shape::long-wings, (214) has wing shape::pointed-wings, (77) has tail shape::fan-shaped tail, (213) has wing shape::rounded-wings, (79) has tail shape::squared tail, (83) has upper tail color::purple, (82) has upper tail color::iridescent, (75) has tail shape::rounded tail