Structured Learning of Compositional Sequential Interventions

Summary

Estimating the causal effect of a sequence of interventions on another sequence is a central task in causal inference / treatment effect estimation. However, for large discrete spaces, canonical assumptions such as Markovian assumptions, short sequences, and so forth will not apply, while very general black-box models may present poor generalizability due to the sparsity of the observed treatment sequences. This work presents a model where the effect of a sequence of interventions on the target sequence can be decomposed as a product of individual time-point-wise effects. This parameterization is fairly general as it parameterizes functions of a specific form exactly, and any measurable function approximately. Identification results are given for the parameter of the proposed parameterization. This motivates a VAE-based fitting approach, together with predictive intervals coming from conformal prediction. Experiments are given, showing competitive performance of the method.

Strengths

The work is original as it shows that coming with a specialized parameterization can help, both theoretically (identifiabiity) and empirically (better performance), than a general black-box model. It is also significant as the parameterization is actually not particularly restrictive, as Propositions 1 and 2 show that it can recover exactly or approximately general types of functions. Thus the method is generally very principled and should be very applicable in the field.

Weaknesses

1) IMO the main drawback of the work is in clarity. 1a) It took me significant time to parse what the field is missing and what is the exact contribution of the method. The contribution is specified in the "Scope" paragraph l.37-42 and later in l.64-78, and the previous work with its drawbacks is scattered in l.43-63, Section 4 l.304-322 and Appendix A. I think that these sections should be completely re-organized, notably all the previous work should be put together and the contributions outlined clearly in contrast to such previous work ; especially as the contribution consists actually in specifying a more restricted functional form that's more tailored to a specific type of intervention sequence. 1b) It is not clear what the exact estimand is, or what the exact estimands are, as they are not stated in the problem statement l.79-90. The functional form $\mathbb{E}[X_n^t(d_n^{1:t}) | x_n^{1:t-1}(d_n^{1:t-1}), z_n]$ is also confusing, as it suggests that it does not depend on covariate values but only on $n$, $t$ and interventions, while from l.173 it also depends on covariates. It is only in Section 4 that I understood that I understood that only predictions were to be made conditional on previous covariates. Thus, I would be happy if authors could state in their rebuttal what estimand(s) are we looking for, and if they could write it here. 1c) It is also not clear how the functional form of Equation 1 incorporates "compositionality" of interventions. While it can be guessed from Equation 2, it is only in l.184-192 that it becomes clear this functional form for $\psi$ incorporates compositionality (if I understood it well?) 2) While experiments generally suggest the relevance of the method, I'd have a few concerns : 2a) There seems to be rather few baselines : GRU-0 and GRU-1 seem rather redundant as they look like mutilated versions of the GRU-2 baseline ; the latter seems like the only actually fair comparison against CSI-VAE as only that baseline takes the whole intervention history into account. The bottom part of Figure 2 indirectly confirms this as only GRU-2 is evaluated. Thus, all of this might leave GRU-2 as the only real baseline. Could another one be implemented? 2b) Conformal prediction is incorporated in the paper, but it is only evaluated in the Appendix and only on the method. Could an analysis include baselines and be moved to the main part? Note : I am aware that it might not easy to obtain new empirical results in the rebuttal period, I am happy to discuss all of this with other reviewers and ACs

Questions

1) I struggle to understand what these mean, can you develop a bit? a) l.38-39 : "including the bursts of non-stationarity that the application of an intervention brings in" b) l.106-108 : "We do not explicitly condition on $D_{1:t}^n$ in most of our notation, adopting a convention where potential outcome indices $D_{1:t}^n$ are always in agreement with the (implicit) corresponding observed $D_{1:t}^n$". 2) l.173 : how exactly does $\mathbb{E}[X_n^t(d_n^{1:t}) | x_n^{1:t-1}(d_n^{1:t-1}), z_n]$ depend on $x^{1:T}$ exactly? Which components of the latter are used? 3) How stringent are Assumptions 3 and 4? 4) Section 3.1 : a) Is identifiability preserved with variational inference in CSI-VAE? b) What is $\mu$ in the equation of l.250, and how does this incorporate the functional form of Equation 1?

Rating

7

Confidence

2

Soundness

3

Presentation

2

Contribution

3

Limitations

Limitations have a dedicated paragraph, and social impact is no different than any other work on treatment effect estimation more broadly. **EDIT (2024/08/11)** : increased my score following authors' rebuttal and discussion.

Summary

This paper considers a special case of a series of interventions where the interventions are categorical and sparse. And interventions can effect later timestamp. This is a form of causal extrapolation. Authors propose to study this using a conditional mean model that utilizes basis functions and subsequently develop corresponding algorithms.

Strengths

1. The problem is interesting and well-motivated. 2. As far as I can tell, the theory is sound.

Weaknesses

1. Section 2.1 is really hard to parse. 2. The assumptions in section 2.2 is not explained intuitively and I don’t know how necessary they are. 3. The experiments are synthetic and semi-synthetic. Although since the nature of the paper is mostly theoretically, I wouldn’t consider that to be a huge issue.

Questions

1. In equation (1), do you need the lower case x to be a potential outcome? 2. Is the notation changed in 2.1? f does not appear in equation (1). 3. For CSI-VAE, why use GRU instead of more modern arch like attention or even LSTM?

Rating

5

Confidence

3

Soundness

3

Presentation

2

Contribution

2

Limitations

The only limitation is how practical these methods are.

Summary

This paper studies treatment effect estimation under sequential, discrete interventions. It is assumed that all treatments are fully independent and impact all future outcomes. It studies the problem of "causal extrapolation" where some combinations of sequential interventions may not have been observed during training. Parametric assumptions on the causal mechanisms are made and used for generalising to unseen interventions. A matrix factorisation approach is developed to fill in the gaps for the unseen interventions. The approach is empirically tested on synthetic and semi-synthetic data.

Strengths

- This paper studies an important and, in my opinion, an understudied problem in causality: causal extrapolation, as it is called in the paper. Many papers look at assumptions that can be made on the causal structure alone, whereas here, the functional form of the structural equations is considered as well. - The paper studies a way to generalize over unseen combinations of interventions. That is an important concern, especially in the time series context.

Weaknesses

- The mathematical formalisation is very difficult to follow. Wild statements are made without properly introducing the objects, motivating the statement or clarifying what assumptions are made in order for the statement to hold (see questions below). It is unclear which parts follow from the assumption of the causal relationships in Fig. 1 and what additional assumptions or approximations are made. - The assumptions on the data generating process are too restrictive for the method to have much hope of being applied to real data (see limitations). **Minor:** - L21 and L 86: Typo "an unit"

Questions

- Eq. 1: Where does this come from? Can any treatment effect that follows the causal structure in Fig. 1 be expressed like this? Or is this a simplifying modelling assumption? - L136: What are ANOVA models? - L123: What does the last sentence in this paragraph mean? - Equation in L135: What does this function correspond to? The equation seems not to hold for general functions: e.g. if the dimension of the two RHS functions is 1, then this is certainly not a statement that is true in general. Under what conditions can you decompose a function like this? What are the terms on the RHS? How do they fit into the initial problem setting? - Prop. 3: Does this assume that equation 1 holds?

Rating

3

Confidence

4

Soundness

2

Presentation

1

Contribution

2

Limitations

- Fig. 1: The assumed causal structure severely limits the applicability of the presented approach. There is no allowed confounding and, more importantly, the interventions are assumed to be independent. The latter assumption would require full randomisation during data generation. It does simplify things a lot, since it avoids having open paths through conditioning on colliders. But, in essence, it is assumed that the data comes from an RCT. Therefore, the contribution is applicable to RCTs that do not cover the full range of combinatorial interventions. - The identifiability Assumption 1 means that we essentially have full access to probe each unit (we need to have the "no intervention" applied for several time steps). That is, we essentially can perform an RCT on all units. In that case, why not keep the units in the lab, rather than predicting what will happen under future interventions?

Summary

The authors propose CSI-VAE, a method that solves the problem of forecasting potential outcomes in multiple interventions. In particular, the authors seem to propose a solution to the problem of a large (combinatorial) space of future treatment plans using the controlled past treatment sequences for inference.

Strengths

STRENGTHS * The paper is well structured, written, and generally easy to follow. * Including an automated (distribution free) uncertainty quantification using CP is a nice to have and will definitely improve adoption into practice. in particular in medical applications * Focusing on identification in this tricky area is very welcome and I would encourage other authors to also include at least a discussion on identification, thank you for this

Weaknesses

WEAKNESSES * The problem sounds fairly general which would allow other treatment effects over time models to also be relevant to discuss and in particular benchmark against. Currently, it seems the authors only benchmark against ablations of CSI-VAE and naive methods using GRU? Am I mistaken?

Questions

I find the paper to be quite well done, my only remark is wrt the benchmark settings which I would like to see addressed in a rebuttal.

Rating

5

Confidence

3

Soundness

3

Presentation

3

Contribution

2

Limitations

see above

Answer to Aug 7 rebuttal

Many thanks to authors for the extensive rebuttal. My questions have been answered, except the following points : 1) I still do not understand exactly how $E[X_n^t(d_n^{1:t})|x_n^{1:t−1}(d_n^{1:t−1}),z_n]$ in Equation 1 and $\phi_l(x_n^{1:t−1}(d_n^{1:t−1}), z_n)$ in Equation 109 depend on $x_n^{1:t−1}, d_n^{1:t−1}, z_n$, or more precisely what the notation $x_n^{1:t−1}(d_n^{1:t−1})$ means. It suggests that $x_n^{1:t−1}$ is a function of $d_n^{1:t−1}$, while of course also depending on $n$ and $t$. Further, from this interpretation, it is at first unclear whether this function is known a priori (before observing the data) or pre-specified, or it is observed in the data, or it has to be learnt, etc... This is what I mean in the statement raised by your own question "we have one question for the reviewer: we didn’t quite get what the reviewer meant by" : if $x_n^{1:t−1}$ is a function that is pre-specified or known a priori, then indeed $x_n^{1:t−1}(d_n^{1:t−1})$ only depends on $d_n^{1:t−1}$ in the argument, and $n,t$ in the index. It is later in the paper that I understood that these $x_n^{1:t−1}$ are actually values taken by the (potential) covariates, thus these $x_n^{1:t−1}$ are vectors or scalars but not functions (if I am not mistaken). Thus, it now seems to me that the $x_n^{1:t−1}(d_n^{1:t−1})$ notation for small $x$ is incorrect and should be scraped altogether and replaced with either $X_n^{1:t−1}(d_n^{1:t−1})$ with capita $X$ in statistical quantities, and $x_n^{1:t−1}, d_n^{1:t−1}$ in functions. To be more precise, I understand that one should write a) $E[X_n^t(d_n^{1:t})|X_n^{1:t−1}(d_n^{1:t−1}) = x_n^{1:t−1}, z_n]$ in Equation 1, which makes it clear that $X_n^{1:t−1}(d_n^{1:t−1})$ is a potential covariate that is indexed at the observed intervention $d_n^{1:t−1}$ and is here taking the observed value $x_n^{1:t−1}$ ; b) $\phi_l(x_n^{1:t−1}, d_n^{1:t−1}, z_n)$ in Equation 109, as this is a function as in l.173 and it does not formally rely on random variables, including potential covariates. Is all of this correct? If not, is $x_n^{1:t−1}$ for a small $x$ actually a function of $d_n^{1:t−1}$? 2) My mistake, I had misread Section 3.1 and missed the point that the mean and conditional variance of l.250-251 are actually the posterior parameters of $\beta_n$ (feel free to correct me if I am wrong here). Here are some questions, mostly for clarity: a) Are these parameters for the encoder then? b) By the conditional mean and variance of the decoder, do you refer to Equation 233? c) Do you also confirm that the VAE departs from the setup of Theorem 1, as notably the basis function is learnt?

Closing comments

Thanks again for your review and we hope we have addressed the question about benchmarks - including the easiness by which further comparisons can be added to our codebase.