Paper: Predictive Coding beyond Correlations (ICLR-2024)

Reviewer comment (Theory):
Page 3: it is noted that the way of removing impact of latent confounder is to do intervention on a randomly selected individual. Then with the intervened data, could we simply do the standard ATE calculation? What is the main advantage of PC model on causal inference compared with other methods?

Author response:
While our method sometimes performs better than existing ones, the main goal of our work is not to solve open problems in the causal inference literature, but to provide a bridge between predictive coding (an influential theory in neuroscience), and the field of causal inference. Hence, we foresee this work to have an impact in the predictive coding community, as it extends the toolbox of computational neuroscientists to contain techniques that allow them to perform causal inference and structure learning, which was never the case before: So far if people wanted to use predictive coding models to perform causal inference, they would have to train a new model for every intervention variable (because it is a different graph after mutilation/intervention). however, we show how the original graph can be modified in the causal inference step in order to render equivalent causal conclusion without the need to train a new model for every intervention variable

Reviewer comment (Theory):
Page 6, classification: Could you further explain the relationship between classification and causal inference under the predictive coding model setting? What are the classes here? And how is test accuracy related to causal effect?

Author response:
The classes are the labels of the MNIST dataset.  The reason why interventions improve the test accuracy on classification tasks is that they prevent the error of the inputs to spread in the network, and hence enforce a specific direction of the information flow, which goes from cause (the image) to effect (the label).

Reviewer comment (Theory):
If the graph is unknown, could the PC model still do causal inference by using the structure learned by PC from observational data?

Author response:
Yes and this can be done without changing the model parameters or the architecture. Thus one could see it as an end to end causal inference engine as shown in our experiments of Appendix H. All that is required is to set the error node of the intervention variable to zero and then perform inference.

Reviewer comment (Novelty):
It is not very obvious how the proposed method compares with the existing frameworks. So, it is also not very clear how the proposed method would contribute to the research direction. An in-depth discussion comparing the PC graph with DAG might lend more weight to the paper's influence on the causality.

Author response:
> How can it contribute to the research direction?

The main goal of our work is not to solve open problems in the causal inference literature, but to provide a bridge between predictive coding (an influential theory in neuroscience), and the field of causal inference. Hence, we foresee this work to have an impact in the predictive coding community, as it extends the toolbox of computational neuroscientists to contain techniques that allow them to perform causal inference and structure learning, which was never the case before. 

The assumptions that we make are coherent with this line of argumentation. For example, we assume causal sufficiency, and we only compare against the large amount of works in causal inference that make this assumption. This is quite common in a branch of the causal inference literature (e.g., Geffner Thomas, et al. "Deep end-to-end causal inference, 2022).

Reviewer comment (Writing):
In equation 1, is the variable $x$ sampled from the standard normal distribution or a parameter of the model?

Author response:
> In equation 1, is the variable $x$ sampled from the standard normal distribution or a parameter of the model?

It is a latent variable of the model, optimized via gradient descent.  We initialize it via sampling. However, the best way this parameter can be initialized is currently an open question.

Reviewer comment (Evaluation):
Is there any intuition of how the PC graph outperforms the existing method? Is it mainly because the proposed model better estimates the observational probabilities?

Author response:
> Is there any intuition of how the PC graph outperforms the existing method? Is it mainly because the proposed model better estimates the observational probabilities?

Yes: We believe that the flexibility (and generality) of our method made by hierarchies of Gaussians, but optimized locally, allows to better estimate the observational probabilities.

Reviewer comment (Presentation):
Figures like fig 4, fig 5, and fig 6 have too much information from too many results. These results can also be presented in much cleaner formats. For example, in structure learning the mean squared error of the learned and actual adjacency averaged over multiple runs would be much more informative than simply showing the adjacency for two graphs.

Author response:
This is merely for a space reason: we have decided to report single metrics in the supplementary material, as every table is as large as a single page.

Reviewer comment (Experiments):
Intervention query — am I correct in assuming that for each intervention query, a new PC graph has to be trained from scratch? I believe this is the case as a different error has to be zeroed out. If so, is this not very inefficient? This is not the case if interventional queries are found by doing graph mutilations.

Author response:
This is also not true: once the model is trained on a specific dataset,, it is possible to perform all the interventions needed, as they do not interfere with the parameters learned.

Reviewer comment (Writing):
Classification task — a lot more detail is needed for this to be clear. It is simply stated that "we perform an intervention on the input" but it is not clear what this means. An intervention in a causal sense means putting in another input which will change the label as well. If you mean zeroing out the errors of the input nodes, this needs to be made clear. "Intervention" has a very specific meaning in causal literature. The Appendix has insufficient details.

Author response:
We will state this more clearly in a future draft of the manuscript. However, performing an intervention on the input simply means fixing the input nodes to the input values, and zeroing out their errors.

Reviewer comment (Experiments):
Structure learning — has the data been normalised? If not, this should be mentioned and compared against standardised data in line with [1].

Author response:
The data has not been normalized. We will add this explanation.

Reviewer comment (Presentation):
Took a while to find where the non-linear data generation was, as Appendix D is never referred to in any of the text.

Author response:
Thank you for all the comments and pointers about the presentation of our work: We will address them all, and they will largely improve the manuscript. We had removed most of the pointers from the main body of the manuscript to the supplementary material, to not refer to the supplementary material in every paragraph. However, at the beginning of the supplementary material, there is a detailed index, divided in the same way as the sections of the main paper, that states where to find the experiments of every topic.

Reviewer comment (Theory):
How does predictive coding handle a single dataset? All the theory in the paper assumes access to a single datapoint.

Author response:
As every other deep learning method: training can be achieved via SGD applied on mini (or full)-batches to minimise the average energy F.

Reviewer comment (Writing):
Section 2.1 — what is time step t referring to? It has not been defined.

Author response:
We will update the description to clarify this. It refers to a single iteration of gradient descent on the value nodes x.

Reviewer comment (Theory):
Given that each x_i is a node in the (causal) graph, how is it possible to use deeper neural networks (eq 1) to learn the relationship between observed variables? The current formulation seems to limit the relationship between variables to 1 layer NNs.

Author response:
Using hierarchical generative models. This is how we have performed the experiments on the colored MNIST dataset.

Reviewer comment (Theory):
Paper is very informally written and theorems / definitions are not up to standards of a statistics community. Eg Theorem 1 has no assumptions stated, what are the spaces allowed of the variables involved etc.

Author response:
>Paper  is very informally written and theorems / definitions are not up to standards of a statistics community. 

The assumptions are stated in section 2, and referred before the theorem, i.e. the distributions we are referring to are the gaussians approximated with dirac deltas. We did not re-state them at the beginning of the theorem due to a lack of space, and avoid repetitions.

Reviewer comment (Novelty):
I don't see actual methodological novelty wrt to causal inference and/or discovery except "interpretations" of existing techniques in the parlance of predictive coding. If the interpretation is the only contribution, can you clarify what is the value here?

Author response:
> I don't see actual methodological novelty wrt to causal inference and/or discovery

The main goal of our work is not to solve open problems in causal inference and/or discovery, but to provide a bridge between predictive coding (an influential theory in neuroscience), and the field of causal inference. Hence, we foresee this work to have an impact in the predictive coding community, as it extends the toolbox of computational neuroscientists to contain techniques that allow them to perform causal inference and structure learning, which was never the case before. Hence, we would like our work to be evaluated in terms of how impactful our findings are for the computational neuroscientist, and for the machine learning researcher that is exploring training algorithms that are alternative to backpropagation, rather than how well our method is suited to solve open problems in the causal inference literature (which is beyond the scope of this work). This is why we have selected 
 applications to neuroscience & cognitive science as a primary area of our work.

Reviewer comment (Novelty):
Theorem 1 is irrelevant and not new or a contribution, I am not sure why it is there. The claim right before it "This approach obviates the need for explicit adjustment formulas and back-door criteria in causal inference." is misleading and hints at a contribution where there isn't one. If I understand correctly, the claim is true simply as an immediate consequence of the "truncated factorization" (Pearl 2009, Eq 1.37) when no hidden confounders are involved. In this case, there is no need for back-door adjustments or other similar things already by standard causal inference.

Author response:
> Theorem 1 is irrelevant and not new or a contribution:

The reason we believe the theorem is relevant is because it formally claims that performing causal inference on a mutilated predictive coding graph is equivalent to performing regular inference on the original predictive coding graph after setting the error node of the intervention variable to zero.

>  The claim right before it "This approach obviates the need for explicit adjustment formulas and back-door criteria in causal inference." is misleading 

Our intention with the statement was in reference to the exchangeability assumption and how it allows us to identify a causal effect. Exchangeability assumes that the graph is in a mutilated form (which is not the case for predictive coding networks because we never perform such a graph mutilation, and keep the original predictive coding graph). Therefore, in classical causal inference one has to adjust for the observed confounder in order to achieve conditional exchangeability. However, in our case it suffices to set the error node of the intervention variable to zero in order to render the mutilated graph in which exchangeability holds again. To ensure clarity and accuracy, we will revise our statement to reflect that our method’s utility is in its application to scenarios with observed confounders

Reviewer comment (Theory):
What are the assumptions in the part before 2.1 "Posterior distribution"? The method is presented in the intro/abstract as very general but here we seem to require Gaussian assumptions, unit variances, mean field approximations .. all these in text and not presented properly mathematically (see comment above too). Is the setting the same as Peters et al (see below reference)?

Author response:
> What are the assumptions in the part before 2.1 "Posterior distribution" ? 

Section 2 is a summary of the framework of predictive coding as defined in the referenced works. Consequently, we make use of the same assumptions to arrive at Eq. 2 and algorithm 1. Going through the details of derivations would take us the whole manuscript, and hence we refer to the many works that have defined it.  Regarding the assumptions you mentioned, they are standard in predictive coding but not limiting (for a reference, see for example [5], [6]). Here, we show that the standard framework is sufficient to perform in the proposed benchmarks. In particular, the assumptions listed in this section, refer only to the underlying probabilistic assumptions used to describe predictive coding under the lens of variational inference and derive eq. 2. 
Regarding the question if our setting is the same as Peters et al. [2018], the answer is no as their work focuses solely on learning the underlying directed acyclic graph with linear functions. Our setting is different as we experiment with both linear and nonlinear functions as well as structure learning and causal inference. Furthermore, we don’t claim full identifiability, instead our method of structure learning identifies the graph from the joint distribution only up to Markov equivalence class assuming faithfulness instead of weak faithfulness (causal minimality).

Reviewer comment (Theory):
At start of 2.1 you say you will use SGD for optimizing (I guess F from Eq 2) - why SGD? This is a simple least squares objective to optimize and simpler (faster) alternatives exist. Unless I'm missing something but again not very formally presented, there is no clear optimization problem stated with argmins etc.

Author response:
> At start of 2.1 you say you will use SGD for optimizing ( I guess F from Eq 2) - why SGD ? 

Eq. 2 is a complex non-convex function and its optimization is currently an open and quite active area of research (see [3], [2]). Currently stochastic gradient descent is the go-to method to optimise such models.  In terms of the optimization problem, our goal is to learn the set of weights W that allow us to compute the posterior p(X_unk|X_data = S_data) for any given S_data. This is achieved, as explained in section 2, by minimising the energy F defined in eq. 2 with respect to the node values X and the weights W as described in algorithm 1. Formally, during training, we are looking for W*, defined as:

X* = argmin_{X}F(X,W)
W* = argmin_{W}F(X*,W)

Where X is the training data and W the set of weights defining the PCN. This is standard in PC literature (see for example [1], section 2.1) and the optimization problem is commonly tackled with a combination of Expectation-Maximization and SGD.