Counterfactual Realizability

We thank the reviewer for the great suggestions and questions! We also appreciate the acknowledgment that the result is novel and important to multiple applications. We address your questions below.

(Q1) Applications: we appreciate the opportunity to clarify a slight misunderstanding here. An output $Y$ cannot be generated multiple times for the same unit under different randomized values (as formally highlighted by Cor. 3.8). Rather, ctf-rand( ) can be used to measure the $Y$ for different units, each generated under a different randomized condition. In practice, $Y$ does not need to be a digital mechanism. In Example 3, $Y$ indicates a user successfully staying within her time limit (each day is a unit). In Example 2, $Y, Z$ could also be independent human evaluators who separately receive hardcopy CVs (each CV is a unit that can be ctf-randomized with different names at the top). Indeed, the field experiment cited in Example 2 involved real-world job applications [1]. The applicability of this technique is quite broad. Still, as noted in Sec. 5, most systems and interactions are being digitized, including customer service, financial transactions, remote working, and recruiting, especially with LLMs entering the ecosystem. This topic is only starting to be recognized, and we expect it to become increasingly relevant and offer other possibilities for new types of experimental design that allow one to access highly informative counterfactual information. Formally, the main condition that allows/precludes one from performing this procedure in such settings is based on the construct we call “ctf-mediator”, as elaborated in the next bullet.

(Q2) Mediator condition: A ctf-mediator $W$ can receive an exogenous noise variable, but the inverse from $W → X$ needs to be deterministic. This means that the domain of $W$ divides into equivalence classes of values that can each be mapped back to a unique $x$ (explained in lines [1724-1727]). This is quite intuitive and aligns with real-life situations. For instance, in Example 2, a race $X=x$ could be associated with many stereotypical names {$\{W=w\}$}, but for each stereotypical name that is chosen for the experiment (e.g. “Nguyen”), it maps back to a unique race (“Asian”). ctf-rand( ) requires that the outcome variable perceives $X$ via some interface channel that can be manipulated, not necessarily digitally. E.g., the hardcopy CV example earlier, or a market-research study where a participant could be made to perceive different signals by altering the physical study environment.

(Q3) Experiments: for Example 2, we describe the intuition for the results in Sec. E.2.2 using a worked numerical SCM where the L2 metric suggests zero discrimination, but the L3 metric is actually non-zero. To ensure we didn’t cherry-pick the numbers, we ran 1,000 optimizations with random seed, which revealed that a divergence between the L2 and L3 metrics occurred in most cases (Fig. 5). This makes sense because L2 is just less informative about the underlying mechanics of the system, and our ctf-randomization procedure is needed to probe L3. For Example 3, we provide a detailed interpretation of the latent variables and mechanisms in lines [2160-2175] to ensure the graph is not arbitrary. The intuition for why L3 performs better is because with each additional ctf-rand( ), we are getting more information about the latent variables $U_1, U_2, U_3$ that affect $Y$. Interestingly, we can even use a descendant of $X$, $D$ to obtain information about $Y$ via the confounder $U_3$. This insight is not present in any previous work, to the best of our knowledge!!

(Q4) Algo walk-through: We provide step-by-step examples for the algorithm in Appendix B.4. [NEW] We have also added a detailed walk-through for the algorithm and proof strategy in Appendix B.2. To summarise: for each variable $V$ in the input graph, we check what are the necessary and sufficient interventions (or lack of interventions) one needs to perform w.r.t each potential outcome $W_{\mathbf{t}}$ in the input query $\mathbf{W}_\star$. This is what the inner loops are doing -- tracking all the required conditions in topological order. If there is no conflict across these conditions collectively, and if the feasible action set contains the necessary actions, the query is realizable. Otherwise not.

We are happy to explain any aspect in more detail! We also explain in Appendix B.2 that rejection sampling was used to keep the presentation clean instead of overlaying concentration guarantees, etc., which distracts from the main point.

We appreciate the meaningful discussion and would be happy to provide further clarification.

Thank you!

References:

[1] Bertrand & Mullainathan. 2003. Are Emily and Greg more employable than Lakisha and Jamal? A field experiment on labor market discrimination

Dear reviewer 4AQU,

We were wondering if you had any questions or feedback regarding the algorithm’s walk-through and proof strategy, and the scope of applications of our method.

We are keen to submit the best version of our paper, and would gladly incorporate further suggestions.

Thank you!

Authors of submission 12927

We greatly appreciate the reviewer’s effort in providing feedback, and do sympathize with their constraints during the review cycle. We sincerely hope to use the extended discussion period to resolve any gaps in understanding the main result.

We hasten to reassure the reviewer that all appendix material is not needed to understand the main results! As we emphasized throughout the appendices (e.g., at the start of Appendices D and E, lines [1677], [2003], [202]), most of the appendix is targeted at the interested reader: detailed case studies for some compelling causal queries (Appendix B.4), pre-empting FAQs (Appendix D), and discussing novel aspects of experiments (Appendix E), which are common at venues like ICLR.

We believe these discussions would improve the impact of our work, since a diverse audience might be curious about various aspects of the results. For instance, industry practitioners at ICLR may want to learn how to actually implement this new experiment-design technique, causality researchers may be interested in the nuances of the causal hierarchy, and applied ML researchers might be interested in developing new methods in fairness or RL. However, the self-contained results in Sec. 3 can be understood without this further elaboration.

Focusing on our main results in Sec. 3, and to address the feedback that it is dense, we tried to provide an increasingly refined view of the results as follows:

(1) An intuitive example illustrating the concept: lines [263-300] + Fig. 3

(2) [UPDATED] A general summary of the algorithm and proof-sketch: lines [301-306, 790-795]

(3) [UPDATED] A detailed walkthrough of the algorithm and proof-sketch: [796-859]

(4) Full proof of the algorithm and theorems: Appendix C

(5) Multiple examples (query + graphs) of applying the algorithm: Appendix B.4

We believe this strikes a reasonable balance between friendliness/legibility, and also depth, for the readers who may be interested in the details. The audience at ICLR is broad, so our strategy was to provide this stratified view of our results. Still, we are certainly open to improving and reorganizing some parts of the paper, if the reviewers think this would be suitable, since all the content is already there!

Specifically regarding Step ii.b., we have updated Sec. B.2 with a detailed explanation of the necessary and sufficient steps and the minimal intervention needed (changes in blue). The induction argument of why it is enough to check conflicts in topological order is given in the full proof.

Besides step ii.b, are there any details not clear?

We remain fully committed to resolving all gaps in understanding!

Thank you for your engagement!

Dear reviewer 4AQU, as the discussion period is drawing to a close, we would like to check if there are any further questions that we can address in our final response.

Thank you!

EDIT: We wish to highlight (for the benefit of the Meta-Reviewer) that we have fully addressed all the clarifications requested by reviewer 4AQU in our submission.

Some of these clarifications are not material to the correctness of our main result (e.g., why we used rejection sampling), and some of these clarifications were already present in the original submission’s main body (e.g. what are the necessary and sufficient actions was described in the original submission in Sec. 3 lines [299-306]), or discussed in the proof.

Still, the reviewer’s suggestion of providing a more detailed walk-through of the algorithm was a good one, and we have incorporated this into the final version of our submission in Appendix B.2.

We thank doTz for their meticulous review and feedback. We are heartened by your acknowledgment of our contribution and its implications.

On the fundamental constraint of experimentation (FCE):

This is an excellent point and gets to the motivation of the FCE assumption. There are two possible settings one could discuss, namely: 1. The full model of the environment (SCM) and the latent variables $\mathbf{U}$ per unit are known; and 2. The SCM and the unit’s identity are unknown.

The first setting involves cases like running simulations, where the experimenter has full access and control over the environment. It could also involve working with an AI agent where you can observe and freeze all model parameters (including noise terms) and re-run inference with different inputs. This seems to be what you describe re: multiple copies. In this setting, we don’t actually need many causal inference tools like identification, and we could just compute values directly from the SCM (in a flavor similar to the semantic evaluation given in Sec. 1.3 in [1]).

In our work, we study the second setting (see also Assumption C.1). In Example 1, this might mean $Y$ is a closed-source AI agent with some temperature setting generating noise terms $U_Y$ in each round. For the same input, different results could be obtained from repeated attempts on the same agent. I.e., we wouldn’t be able to freeze the random seed and evaluate the outcome under doctored yellow-color and red-color. For a single round, we are limited to just one set of inputs $W=yellow$. Repeating mechanism $Y$ again with $W=red$ means the unknown $U_Y$ term will have changed. In this case, this does not match the counterfactual definition, since the quantities are not being scoped relative to the same unit.

Importantly, this second setting realistically represents how we will interact with AI agents and many other interactions in the real world when the underlying mechanisms (or human behavior) are unknown. For instance, in Example 1, the experimenter is an auditor making API calls to a closed-source proprietary system they cannot control. Also note: even the so-called fundamental problem of causal inference (Cor. 3.8 and [2]) wouldn’t arise under your setting of freezing a model.

Minor comments:

• Line [111]: Yes, there was a typo! $\mathbf{W}_*$ is a random vector containing multiple potential outcomes, and $\mathbf{w}$ is a specific vector value drawn from the support of the former. The external summation is over units $\mathbf{U}=\mathbf{u}$ that would deterministically induce the random vector to take the value $\mathbf{w}$. The inner product of indicators should check whether each potential outcome $W_t$ in the random vector takes the required value in $\mathbf{w}$ - this is corrected now!

• Def. 2.1: The way we think about it, $f_V$ is the mechanism which, you are right, is not observed in general. It produces some observable feature $V$ for the unit. And READ is the action of recording this observed value. We have reworded slightly to reflect this.

• Confounder in Fig. 2. Also makes sense!

References:

[1] Bareinboim et al. 2022. On Pearl’s Hierarchy and the foundations of causal inference

[2] Holland. 1986. Statistics and Causal Inference

And I deeply appreciate you allowing me to learn from you and update my conventional belief.

All questions are addressed and I have adjusted my score accordingly.

We thank the reviewer for the insightful comments. We are heartened by your interest in this topic! We believe this is an important result for the field of causal inference, and are optimistic that the above concerns can be successfully addressed.

Your main questions are (pls correct us if we missed anything!) : what is/are

(a) some intuitive and formal reasons why counterfactuals are believed to be non-realizable by definition?

(b) some references pointing to this conventional belief?

(c) bounding, and related references?; and

(d) some topics we mean by “causal machine learning”, so that you can place our work in context?

We appreciate the candor that you are not as familiar with the subject.

To respond:

(a) A counterfactual distribution is some $P(W_*)$ where $W_*$ is a tuple of potential responses from different regimes (we explain this in Preliminaries; also see Fig. 1.13(iii) in [1]). E.g., suppose $X$ is a drug dosage and $Y$ is a health outcome. The joint distribution $P(Y_x, Y_{x’}, X)$ means we are reasoning about the probability that $Y$ would have taken some value $y$ had we done $do(x)$ for a patient in a randomized trial, and $Y$ would have taken some value $y’$ had we instead done $do(x’)$ for the same patient, and the patient would naturally have been assigned $X=x’’$ in the absence of any randomized intervention. By definition, these scenarios evoke “parallel worlds”. For any given patient, since we can only study her under one experimental condition (world), it is generally believed that such counterfactuals cannot be physically realized.

(b) This is actually a really common belief, sometimes taken as conventional wisdom. We have updated in footnote 1 the following quotes from influential researchers: “By definition, one can never observe [counterfactuals], nor assess empirically the validity of any modeling assumptions made about them” [2], and also the quote “...The problem with counterfactuals like [$P(Y_x \mid x’)$] is [that] … we simply cannot perform an experiment where the same person is both given and not given treatment” [3]. But as we explicate in Example B.1, $P(Y_x \mid x’)$ indeed is experimentally realizable, by measuring $Y_x$ and $X$ in the same world (i.e., we can “realize” this without literally entering parallel worlds). Still, the belief has persisted that “counterfactual judgments remain hypothetical, subjective, untestable, unfalsifiable” [4], due to the understanding that parallel worlds are inaccessible. This matters because agents who don’t leverage this capability (believing it impossible based on this limited understanding) can suffer badly, as Examples 2 and 3 illustrate. Note: the other reviewers acknowledge that the formal results are novel.

On a related note, we respectfully disagree re: Assumption 3.1. We feel it is evident that in most real-world cases a physical process, once performed, cannot be reversed without changing underlying features, with a justification sketched in [227-236]. Relaxing this assumption may be less realistic for practitioners.

(c) The main reference is [5] which is now clarified in Sec. 5 lines [505-508]! Bounding is a task within the identification setting. Namely: using causal assumptions such as a causal graph, we can show that a causal query is bounded within some range computable from the available data. If the query is “identifiable” from the input data, the range collapses to an exact value. As discussed in Sec. 5, it stands to reason that if we can get more kinds of data from richer experiments, the bounds can only tighten further.

(d) We mean areas like causal fairness analysis, causal RL, causal generative modeling, and others that leverage the tools of causal inference to improve the performance, trustworthiness, and scope of ML methods. The excellent survey in [6] covers some of these but is slightly dated (it doesn’t cover some recent developments). We highlighted at least three areas in our examples where counterfactual realizability is directly relevant: explainability (mediation), fairness analysis, and reinforcement learning.

Please let us know if this makes sense and if you have further feedback! We are fully committed to resolving all pending issues.

References:

[1] Bareinboim et al. 2022. On Pearl’s Hierarchy and the foundations of causal inference

[2] Dawid. 2000. Causal Inference Without Counterfactuals (with Comments and Rejoinder)

[3] Shpitser & Pearl. 2007. What Counterfactuals Can Be Tested

[4] https://www.inference.vc/causal-inference-3-counterfactuals/

[5] Zhang et a. 2022. Partial counterfactual identification from observational and experimental data

[6] Kaddour et al. 2022. Causal Machine Learning: A Survey and Open Problems

We thank jnLW for the valuable feedback and expertise, and appreciate the opportunity to discuss some of these nuanced aspects of our work!

A few other references are (Geneletti & David, 2010) and the notion of SWIGs (Richardson & Robins, 2013)

Indeed both are very relevant papers! We do cite the latter in Sec. 5 to explain the difference vs. our work. We now added the former in Sec. 5 line [502] as well.

Geneletti & David cast the unit’s natural decision as a separate variable, say $T$, and use identification arguments to compute ETT. This framing does not lend itself to a causal RL algorithm which optimizes $\mathbb{E}[Y_x \mid X=x’]$. In the latter, it is very clear when exploring the arms of the bandit problem that when $x = x’$, you can hot-start the learning using the counterfactual relationship between the variables (e.g. Equation 41 in Appendix E.3.1). If you frame it as $\mathbb{E}[Y_x \mid T=t]$, this would be the “mere” contextual bandit problem we discuss in lines [486-490]. The second reference does not face this issue, and does cast $T$ as the natural decision variable $X$.

Still, this discussion highlights our contribution perfectly. As explained in lines [210-212], ctf-rand( ) in our framing differs from Fisherian rand( ) in two ways: (1) the natural value of $X$ is leveraged, and (2) where possible, counterfactual mediators allow for multiple simultaneous randomizations of $X$, each affecting different subsets of children.

Both Geneletti & David and Richardson & Robins acknowledge (1) but not (2). (2) is not possible in their formalism. Consequently, their approach wouldn’t discover the “optimal L3 strategy” we discovered in Example 3 in Table 1 and Fig. 6(c-d; purple plot), which optimizes $\mathbb{E}[Y_x \mid x’, d_{x’’}]$. They would be limited to the “ETT baseline strategy” that we show in Table 1 and Fig. 6(c-d; red plot), which optimizes $\mathbb{E}[Y_x \mid x’]$. Further, the approach in both references can’t be applied in Examples 1 and 2 either, since these examples require path-specific randomizations. Still, we added the reference as mentioned.

(Q1) Interpreting counterfactual randomization as an L2 intervention on the perception.

This is a great point, and touches on the core issue. We want to clearly establish that the difference between L2 and L3 is not about realizable vs. non-realizable. Rather, L3 is about sampling joint potential outcomes from different regimes.

Yes, we could technically construe ctf-randomization as L2 randomization on perception. But this misses the counterfactual connection between perceived treatment and different potential outcomes via structural constraints. Philosophically, this really is a different kind of randomization procedure we are formalizing.

In a similar vein, we could technically see an L2 distribution like $P(Y; do(x))$ as “just” an L1 distribution, albeit in a different environment that has undergone a “distribution shift”. But this misses the whole connection between $P(Y; do(x))$ and $P(Y \mid x)$ via structural constraints in the SCM. Philosophically, this justified the development of a new L2 semantics that captures that connection.

(Q2) Perhaps the authors could make Remark C.7 an axiom that defines this measure?

Good suggestion, thank you! We have added lines [1206-1214] in Appendix C.1 (Assumptions), to clarify the probability measure of the agent’s belief. Let us know if that makes sense.

We welcome further suggestions for improving the paper, and appreciate the thought-provoking engagement!