Predictive Coding beyond Correlations

We thank the reviewer for its time and effort.

What is the main advantage of PC model on causal inference compared with other methods?

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

Could you further explained the relationship between classification and causal inference under the predictive coding model setting? What are the classes here? And how test accuracy is related to causal effect?

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).

If the graph is unknown, could the PC model still do causal inference by using the structure learned by PC from observational data?

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.

We thank the reviewer for its time and feedback on the manuscript.

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).

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.

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.

We thank the reviewer for its time and comments.

It should be made more clear that the method assumes that the observed variables are Gaussian distributed (eq 1).

Note that this is not the case: It is an assumption of the variables of the model, but it doesn’t need to be true for the input data, otherwise we wouldn’t be able to train a classification network for which the output is a one-hot encoded vector. This is true only for the variables of the model itself, that is an hierarchy of Gaussian.

Am I correct in assuming that for each intervention query, a new PC graph has to be trained from scratch?

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.

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 its not clear what this means.

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.

Has the data been normalised?

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

How does predictive coding handle a single dataset?

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

Section 2.1: What is time step t refering to? It has not been defined.

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

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?

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

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.

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.

pointers on clarity

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.

We thank the reviewer for the comments and the detailed feedback.

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.

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

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).

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.

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.