Causal vs. Anticausal merging of predictors

We thank the reviewer for their comments. We will first give some general comments, then answer each one of the points in the weaknesses and last answer the questions.

We noticed the strengths of the paper are the same as the summary. Is this correct?

Weaknesses:

1. The question about scaling is an important one. We believe that the results of Equations 7 and 10 (and Theorem 5) would remain unchanged whenever we have $D$ experts to merge. We studied the bivariate case because it allowed us to easily analyse and visualise the implications of our theoretical results. Furthermore, we recently got an interesting insight about the results in high dimensions. Intuitively, the result is that using the same data as constraints, the resulting model in the anticausal direction gives more weight, relative to the causal direction, to those variables that have higher covariance with the target variable.

2. We are unaware of other papers that look at merging of predictors in the case of causal and anticausal learning. If the reviewer knows one such analysis we would be happy to compare it to ours. We read [1] in detail but we found that beyond the use of the term pooling of experts there is no strong relation between [1] and our work. Our work studies the asymmetries between merged predictors whenever we can make causal assumptions, whereas [1] proposes a method to do merging of experts in the setup of foundation models with relation to transfer learning, which we do not talk about in our paper. There is no causality involved in [1].

3. We disagree with the reviewer with this statement. CMAXENT has been studied for the causal discovery and merging data scenario Garrido Mejia, et.al. (2022). What we study here is theoretical asymmetries from merging of experts with different causal assumptions. These are different tasks and we chose CMAXENT because it allows us to merge data, and allows us to include causal information on the final model. Furthermore, we analyse the geometry of the decision boundaries of different models using the same data. This is the only study we know that does such analysis a we find it to be of high value for the NeurIPS community.

4. We do not see how the analysis of non-causal scenarios is relevant to the purpose of our paper, which is studying the differences arising from causal assumptions when merging experts. There is previous research, referenced on our paper, that studies merging of experts using MAXENT without any causal assumptions (see references [21], [23] and [30] on our paper for some examples). This is an interesting question but, as the previous one, out of scope for the current work. Furthermore, there is research on the results of MAXENT under data noise, whereas there is very little research on merging of experts and causality.

Questions:

1. Although this is an interesting question, we find it to be out of scope as we mentioned above. In this paper we are interested mainly in the asymmetries arising from the differences between causal questions on merging of experts.

2. As usual in causality, if there is covariance shift one needs to understand whether the model is causal or anticausal, as a distribution shift can be interpreted as an intervention under the causal lens. If the model is causal, then the it is robust against distribution shifts, if the model is anticausal, then the model will not be robust against distribution shifts.

3. We do not assume that the distribution of any covariate is Gaussian. This is a result of MAXENT under mean and (co)variance constraints. Several common distributions are the result of a MAXENT problem under moment constraints: Bernoulli, Exponential, Gaussian, etc. An excellent resource to understand the relation between common distributions and Maximum Entropy is Wainwright and Jordan (2008).

4. If the reviewer is interested in merging of experts without making any causal assumptions they can check references [21], [23] and [30] on the main document and more recently:

• Vetter, J., Moss, G., Schröder, C., Gao, R., & Macke, J. H. (2024). Sourcerer: Sample-based maximum entropy source distribution estimation. arXiv preprint arXiv:2402.07808.

Dear reviewer,

We hope that we were able to address your concerns and clarified some misunderstandings. Let us know if you have any further concerns and, if not, ask to reconsider the score. We appreciate the time and effort you have spent to review our paper.

Thank you, The authors

We thank the reviewer for taking the time to read our paper in detail and appreciate that they ranked the soundness and contribution of the paper as excellent. We would like to answer the reviewer’s concerns outlined in the weakness section, clarify some points that might have not been clear in the article and answer the reviewer’s questions.

Weaknesses:

About literature and privacy: We appreciate the reviewer’s comment on the literature. However, it is almost always the case that one can include more related work. We will include more work on overlapping datasets and admit that our own point of view is biased by the causal setting. We would also like to invite the reviewer to point us to some relevant papers about privacy and federated learning, which we will be happy to study and reference. We were under the impression that federated learning usually does not study the setting where the variable sets differ (i.e., the merging problem), but following your comment, we have been able to find some papers that study “vertical federated learning”, which we shall discuss. Our impression is that this area focuses more on practical applications and less interested in theoretical statements (and they do not connect the problem to the marginal problem of statistics), but it will be interesting to make those connections.

About inferring causal graphs: The reviewer mentions in the weaknesses section that we are framing the merging of experts’ problem as one of inferring causal graphs. This is not entirely correct, and we would like to clarify it here (and we will do it in the revised version of the text as well). What we want is to come up with a single predictor of our target variable $Y$, given a combination of predictors. We have the information of the causal graph, so we don’t need to infer it. We explore the differences in the solutions of the merging of experts under different causal assumptions (in our case, whether the variables are causes or effects of our target variable).

Questions: About literature expansion: We will expand the literature with the overlapping datasets literature, and we are happy to expand it with privacy and federated learning related research.

About an example: This is a fair point. However, we believe that the greatest merit of the paper are its theoretical contributions and not a potential empirical example. Indeed, Figure 2 is such an example for one particular data generation process, where we can clearly see the difference in the slope of the decision boundary for data with the same moments, under different CMAXENT solutions. If the reviewer is curious about the results of a particular example, we would be happy to run those tests and include them in the final version of the paper.

For a potential real world example (which we can include on the main paper) in the medical domain. If $Y$ is the presence/absence of a disease, the causal scenario is given when $X_1,\dots,X_n$ describe risk factors, while $X_1,\dots,X_n$ being symptoms would be anticausal. Research results from different labs may provide predictors of $Y$ from different sets of risk factors and/or different sets of symptoms. Combining them into a joint predictor which uses the union of all features would be highly valuable, but it also requires to include the right causal assumptions: risk factors cause diseases cause symptoms.

About code: We are happy to share the code for our paper upon acceptance. However, we believe that most of the value of our paper comes from the theoretical insights we are giving on the problem of merging of experts when different causal assumptions are made. In fact, the code to produce Figure 2 consists of a very basic data generation process and a common python library to estimate the decision boundaries using the logistic regression and linear discriminant analysis, as Remark 2 and Theorem 5 of our paper point out.

We thank the reviewer for the careful reading of our paper. We also thank the reviewer for pointing that the soundness and the contribution of the paper are good and the presentation is excellent. We will start giving a short comment on the reviewer's summary and start answering to the weaknesses and questions.

Comment about the reviewer’s summary: When we study the difference in the decision boundaries, we study the case where the moments are the same and not different. This is precisely one of the most surprising results of the paper.

Weaknesses: We appreciate the reviewer's comment about the connection between the literature; in the revised version of the paper we will be more explicit about the connections between “On causal and anticausal learning” and some of the more recent OOV literature.

In particular on the paper “On causal and anticausal learning” they study common Machine Learning tasks (distribution shift, semi-supervised learning, transfer learning) under the light of causal assumptions. They answer questions of the type: can we perform this task if the data comes from this assumed -causal- process? In this paper we do the same but with the task of merging of experts. Here we ask: What are the differences of the resulting models, if we assume the predictors are from a causal process versus an anticausal one? We answer this question using CMAXENT because CMAXENT 1) allows merging of predictors, and 2) allows to include causal information. Nevertheless, we hypothesise the resulting asymmetry would hold also for any other model that has these two properties.

In relation to OOV, we give an example of a DGP and a model so that we can do OOV. The OOV literature is fairly recent, in fact “on causal and anticausal learning” does not study an OOV scenario. We consider OOD and OOV "dual" modes of generalization but their precise link is yet to be understood. This is not part of the current papers technical contribution.

We can include an extended discussion of the above in the revised version of the paper.

This is an interesting question and we indeed have the results for the combined DGP but did not include them in the submission to keep it somewhat decluttered. Since the reviewer believes in the value of the combined DGP we can include it in the final version. As a summary of the results, the resulting conditional distribution of $Y$ given $X$ is the product of the two conditional distributions found in Proposition 1 and Theorem 5. This is expected, as given $Y$, the parents and effects of $Y$ are independent and so their joint distribution is the product of their distributions.

Questions: We will improve the introduction and explanation of certain terms in the manuscript, this is valuable feedback. We do understand merging of experts as the task of putting predictors together into a single model. However, the results show that it matters whether you assume that the predictors are produced with causal or anticausal variables. If you can make these assumptions, then a different question arises: What do we want the combined model for? If we need the model to estimate causal effects then we conclude we should not use the assumed anticausal covariates, if it is about prediction, we should merge all the models available.

Typo: Thank you for reading the paper in such a level of detail. We will correct the typo and double check if we missed more.

We thank the reviewer for their comments. We would like to clarify certain points related to the weaknesses and respond to their questions as best we can.

W1 (about high dimensionality): in the examples we studied, we used two covariates but the results would be similar if we had predictors of more covariates. That is, if we had $D$ covariates we would obtain a transformed logistic regression in the form of Equation 7, but instead all the variables for which we have $Cov(Y,X_i)$ would appear on the right-hand side of the equation. Likewise, for the anticausal direction we would obtain the LDA algorithm where the Gaussians are $D$-dimensional as opposed to bivariate Gaussians. We used only two covariates to be able to visualise the differences in the geometry of the decision boundary.

Furthermore, we recently got an interesting insight about the results in high dimensions. Intuitively, the result is that using the same data as constraints, the resulting model in the anticausal direction gives more weight, relative to the causal direction, to those variables that have higher covariance with the target variable.

W1 (about discrete data): there are two options here:

• First, if the target variable is discrete, in which case our results remain unchanged. In fact, Theorem 7 is written for a discrete (and not bivariate) target variable, and thus a generalisation of the previous results.
• Second, the covariates are discrete. If the reviewer is referring to this case, we would agree that the study of discrete covariates requires a separate analysis. However, to showcase the asymmetries between causal and anticausal merging we decided to resort to cases that admit an analytical solution, which result in differences that are easier to interpret.

W2: (about a potential application of the results) Merging predictors will get increasingly relevant for instance in the medical domain. If $Y$ is the presence/absence of a disease, the causal scenario is given when $X_1,\dots,X_n$ describe risk factors, while $X_1,\dots,X_n$ being symptoms would be anticausal. Research results from different labs may provide predictors of $Y$ from different sets of risk factors and/or different sets of symptoms. Combining them into a joint predictor which uses the union of all features would be highly valuable, but it also requires to include the right causal assumptions: risk factors cause diseases cause symptoms.

W3: We know the content of references [1,2] in detail. Neither of them describe a merging scenario. The paper [1] first studied the implication of causal asymmetries (causal vs. anticausal) for machine learning, for the case of prediction (but not merging). This indeed inspired us, as mentioned in the introduction. The relation to [2] is much more remote: that paper develops a theory of causality that uses Kolmogorov complexity rather than statistical (Shannon) information to study conditional independence properties of causal graphs. It is fundamental for the justification of independent causal mechanisms, which in turn is related to machine learning, but we felt that we did not have to cite it. If the reviewer feels otherwise, we are happy to change this, and include a discussion.

Q1: We are not completely sure by what the reviewer means with incomplete datasets. Assuming the reviewer is asking about the case where we do not have predictors for all causal parents, $Y$ (that is, in the causal scenario there may be additional causes). If we do not have predictors for each causal parent of $Y$, then we can still combine the predictors into a single model, however we are not going to obtain an approximation of the causal mechanism from the causal parents of $Y$ to $Y$ (recall the causal mechanism is simply $P(Y\mid PA(Y))$). As a consequence, we might not be able to compute some causal quantities like interventional distributions without any further assumptions. Of course if the goal of merging the experts is purely prediction there is no consequence of not having all the causal parents, however, as we have shown on the paper the geometry of the decision boundary (regardless of whether one wants to predict or perform causal tasks) -will- change depending on the causal assumptions.

Q2: This is an interesting question and it relates to our interpretation of Q1. In particular (in the causal scenario), if we have the sample averages of the cross-moments between the causal parents of $Y$ and $Y$, then CMAXENT gives us an approximation of the causal mechanism of $Y$ that is robust to distribution shifts. To be more precise, Grunwald and Dawid (2004), and Farnia and Tse (2016) prove that MAXENT and conditional MAXENT are robust Bayes in the family of distributions with the predetermined constraints. That is, it is the distribution chosen from the set of distributions that has the same expectations as those given as constraints, so that the expected value of the log loss is minimised in expectation.