Paper:
Equality of Opportunity in Classification: A Causal Approach
The Equalized Odds (for short, EO) is one of the most popular measures of discrimination used in the supervised learning setting. It ascertains fairness through the balance of the misclassification rates (false positive and negative) across the protected groups -e.g., in the context of law enforcement, an African-American defendant who would not commit a future crime will have an equal opportunity of being released, compared to a non-recidivating Caucasian defendant. Despite this noble goal, it has been acknowledged in the literature that statistical tests based on the EO are oblivious to the underlying causal mechanisms that generated the disparity in the first place (Hardt et al. 2016). This leads to a critical disconnect between statistical measures readable from the data and the meaning of discrimination in the legal system, where compelling evidence that the observed disparity is tied to a specific causal process deemed unfair by society is required to characterize discrimination. The goal of this paper is to develop a principled approach to connect the statistical disparities characterized by the EO and the underlying, elusive, and frequently unobserved, causal mechanisms that generated such inequality. We start by introducing a new family of counterfactual measures that allows one to explain the misclassification disparities in terms of the underlying mechanisms in an arbitrary, non-parametric structural causal model. This will, in turn, allow legal and data analysts to interpret currently deployed classifiers through causal lens, linking the statistical disparities found in the data to the corresponding causal processes. Leveraging the new family of counterfactual measures, we develop a learning procedure to construct a classifier that is statistically efficient, interpretable, and compatible with the basic human intuition of fairness. We demonstrate our results through experiments in both real (COMPAS) and synthetic datasets.

Review:
This is a well-written, detailed paper on an important topic. The paper seeks to decompose sources of discrimination into spurious, direct and indirect effects. It strengthens connections between the causal literature on direct effects to the literature on discrimination in classification by connecting to the notion of Equalized Odds.  Specific comments:  (1) It is not clear what is the overall goal here. Other papers e.g. Nabi and Shpitser talk about trying to remove the effects of "unfair" pathways. Are there specific pathways that the authors think are "unfair"? Why does it make sense to condition on y?  This is done in the EO measure, but is that a good idea? Some contrast relating conditional vs. marginal measure would be useful.  (2) There is a close connection between what are called in this paper "conditional" counterfactual effects and papers on the effect of treatment on the treated. Also effects in strata defined by post-treatment outcomes. For example: Marshall M. Joffe (2002) Using information on realized effects to determine prospective causal effects. https://doi.org/10.1111/1467-9868.00311  (3) The authors in some places suggest that the model in Figure 1(b) represents a canonical graph for this problem. This is inaccurate because the SCM here presumes that there are no unobserved variables (so that the U terms are independent). In general there could be unobserved latent variables giving rise to confounding between any pair of variables in X,Z,Y,M. (hat{Y} is not confounded since it is a function of predictors that are, by construction, observed.)  (4) Lines 122-124: The authors say that additional evidence (Y=y) mean that "the original causal diagram no longer captures in an explicit fashion all the underlying mechanisms that relate X and hat{Y}" This seems to be a category mistake: the presence of additional evidence can induce (probabilistic) independence or dependence but it does not change mechanisms!  To address this, the authors suggest modifying the causal diagram (line 126). It is not clear why one couldn't just condition on y and use d-separation/connection as usual. (As noted, it seems a conceptual error to think of information changing a causal process.)  (5) In the definition of conditional causal model why is M_e required. Note that in the subsequent definitions interventions are on x_0 and x_1, while the "evidence" is just x (which could be some other value that is not x_0 or x_1).  Related: why are the potential outcomes in (1), (2), (3) subscripted by "e" (or "y"). Why does it make sense to consider intervening on e or y (since this is a response (that doesn't directly affect hat{Y}. Could this be left out as a subscript / intervention here? (Obviously e or y need to be conditioned on.) I guess the text at lines 144-145 makes the point that these could be omitted - though it seems there is an error here (see below) - but this doesn't explain why they were included in the first place!  (6) Lines 233-236: The authors suggest that direct and indirect effects could be learned from a randomized experiment. Though true of "controlled direct effects", this is not true of "natural" direct and indirect effects  such as (1) and (2) that involve nested counterfactuals y_{x1,.. W_{x0}}.   J. Robins and T.S. Richardson.  (2011).  Alternative Graphical Causal Models and the Identification of Direct Effects.  Chapter 6.  Causality and Psychopathology: Finding the Determinants of Disorders and their Cures, pp. 1-52.   http://www.csss.washington.edu/Papers/wp100.pdf  As noted by Robins & Richardson, the assumption of independent errors in an SCM is not empirically testable via randomized experiments (since it implies independence of cross-world counterfactuals such as W(x0) and Y(x1,w).  Minor comments: line 3 Hartz et al is not in the Bibliography. [Hardt et al??] line 43 functions => function lines 144-145. "Since Y is a non-descendant node of W, hat{Y}, it has no effect in the counterfactuals before the conditioning bar in the factors of Eq.1"  this should be: Since W, hat_{Y} are non-descendants of Y... no effect *on*...  it is also a bit hard to follow because Y doesn't show up in these terms, it seems implicitly e=Y here (?)  line 155: defender => defendant (?) [Unless this is about football players!]  line 170 Equation (3) SE_{x0,x1}. This notation is a bit inconsistent with that for DE and IE because x0 and x1 are not counterfactuals.  line 184 featbure  line 199 may not been  line 208 remove "as"  line 240 and 241. End of clause (1)  and of clause (3). "blocked by by C" in which graph?  line 249. where P(hat{y}_{x,w}). I think this should be: P(hat{y}_{x,w,hat{z}})  line 262 the the  line 274: Theorem 3. The assumption of an SCM with independent errors  is required here. (This is not empirically testable with any randomized experiment.)  line 299 (before) Figure 4 (b) Experiment 2. What do the bars going from 0.1 to -0.1 mean? What is the difference between these and no bar at all?  lines 304-318. Is it the case that the counterfactual conditional contrasts (DE, IE etc.,) are known here because the generating mechanism / SCM is known? (In other words, no estimation is being carried out.)  line 323. smaller equalized *odds*  line 330 line 328 to 330. In the context of this real data example why would one believe that the conditions in Theorem 2 apply (i.e. ctf explainability). Is it reasonable to assume there are no unmeasured confounders?  ==================================================  The authors reply:  "3. Conditional Causal Models (R3) The conditional causal model does not modify the existing causal mechanisms, but only highlights the spurious and causal pathways that are not distinguishable by d-separation in the (standard) original causal model."  This sounds reasonable, but it is *not* in keeping with the text in the original paper. Specifically, lines 122-127 in the original paper say the following:  "This phenomenon exposes that, whenever an additional evidence is provided (Y = y), the original causal diagram no longer captures in an explicit fashion all the underlying mechanisms that relate X and hat{Y} We propose a simple modification of the causal diagram to account for the aforementioned extra mechanisms arising due to conditioning."   My guess is that the word "mechanism" here is the source of the problems. This suggests changing the "underlying" SCM. I think that if the authors replaced this with "active pathways" it would have caused much less confusion.