A comparison of causal scrubbing, causal abstractions, and related methods

6jenny

2Erik Jenner

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This is a nice comparison. I particularly like the images :) and drawing the comparisons setting aside historical accidents.

A few comments that came to mind as I was reading:

Perform an interchange intervention on the treeification of L such that the corresponding intervention in the treeification of H would not change any values.

As far as I saw, you don’t mention how causal scrubbing specifies selecting the interchange intervention (the answer is: preserving the distribution of inputs to nodes in H, see e.g. the Appendix post). I think this is an important point: causal scrubbing provides an opinion on * which* interventions you should do in order to judge your hypothesis, not just

We need some way of turning a neural network into a graph L, i.e. we need to decide what the individual nodes should be. We won’t discuss that problem in this post since it is orthogonal to the main algorithms we're comparing.

I actually think this is reasonably relevant, and is related to treeification. Causal scrubbing encourages writing your graph in whatever way you want: there is no reason to think the “normal” network topology is privileged, e.g. that heads are the right unit of abstraction. For example, in causal scrubbing we frequently split the output of a head in different subspaces, or even write it as computing a function plus an error term.

TBC other methods could also operate on a rewritten, treeified graph, but they don’t encourage it and idk if authors/proponents would endorse.

Treeification is the one way in which causal scrubbing is stricter than all the other methods.

Related to the above comment: I actually don’t think of treefication as making it stricter, rather just more expressive. It allows you to write down a hypotheses from a richer space to reflect what you actually think the network is doing (e.g. head 0 in layer 0 is only relevant for head 5 in layer 1, otherwise it’s unimportant).

Recall that causal scrubbing only allows interventions that don't change any of the values in the explanation H.

IMO this isn’t a fundamental property of causal scrubbing (I agree this isn’t mentioned anywhere, so you’re not wrong in pointing out this difference; but I also want to note which are the deepest differences and which are more of “no one has gotten around to writing up that extension yet”).

Thanks! Mostly agree with your comments.

I actually think this is reasonably relevant, and is related to treeification.

I think any combination of {rewriting, using some canonical form} and {treeification, no treeification} is at least possible, and they all seem sort of reasonable. Do you mean the relation is that both rewriting and treeification give you more expressiveness/more precise hypotheses? If so, I agree for treeification, not sure for rewriting. If we allow literally arbitrary extensional rewrites, then that does increase the number of different hypotheses we can make, but these hypotheses can't be understood as making precise claims about the original computation anymore. I could even see an argument that allowing rewrites in some sense always makes hypotheses less precise, but I feel pretty confused about what rewrites even are given that there might be no canonical topology for the original computation.

Not sure if I'm fully responding to your q but...

there might be no canonical topology for the original computation

This sounds right to me, and overall I mostly think of treeification as just a kind of extensional rewrite (plus adding more inputs).

these hypotheses can't be understood as making precise claims about the original computation anymore

I think of the underlying graph as providing some combination of 1) causal relationships, and 2) smaller pieces to help with search/reasoning, rather than being an object we inherently care about. (It's possibly useful to think of hypotheses more as *making predictions about the behavior** *but idk.)

I do agree that in some applications you might want to restrict which rewrites (including treeification!) are allowed. e.g., in MAD for ELK we might want to make use of the fact that there is a single "diamond" (which may be ~distributed, but not ~duplicated) upstream of all the sensors.

Summary:We explain the similarities and differences between three recent approaches to testing interpretability hypotheses:causal scrubbing, Geiger et al.'s causal abstraction-based method, andlocally consistent abstractions. In particular, we show that all of these methods accept some hypotheses rejected by some of the others.Acknowledgements:Thanks to Dylan Xu and Joyee Chen for many conversations related to this post while they were working on their SPAR project! And thanks to Atticus Geiger, Nora Belrose, and Lawrence Chan for discussions and feedback!## Introduction

An important question for mechanistic interpretability (and other topics) is: what type of thing is a mechanistic explanation of a certain neural network behavior? And what does it mean for such an explanation to be

correct?Recently, several strands of work have (mostly independently) developed similar answers:

causal abstractionsto neural networks since 2020. They interpret both neural networks and potential explanations ascausal modelsand say that an explanation is correct if it is a valid causal abstraction. To test an explanation empirically, they developedinterchange interventions, which perform the same interventions on the neural network’s activations and on the explanation, after which they check that the outputs after intervention are the same.Causal Scrubbingalgorithm to test interpretability hypotheses. Again, computational graphs are used as explanations, and the validity of an explanation is checked by performing interventions (orablations) on activations.abstractions of computationsthat is very similar to causal abstractions (and many other related ideas, like abstract interpretation or bisimulation). We will call the particular definition of abstractions given in that postlocally consistent abstractions (LCAs).In this post, we explain these different approaches and highlight their similarities and differences. We also give examples of "disagreements" where some approaches accept an explanation and others reject it, and we discuss some conceptual takeaways.

This post is meant to be self-contained and doesn't require familiarity with any of the work we are discussing. (In fact, it may be a good first introduction to all these methods.)

The approaches we listed certainly aren’t the only possible definitions of “explanations”, for example

heuristic argumentscould provide a rather different notion. There is also much older work with similar ideas. We focus on these because## Use cases for definitions of explanations

Historically, the approaches we’ll discuss have been used for different purposes (for example, causal scrubbing has mostly been used to check interpretability hypotheses, whereas Geiger et al. have also been working on interchange intervention training, allowing them to enforce certain structures onto neural networks).

But this is mostly historical accident, so we want to clarify that these methods (and many others) can all be used for several different purposes:

## Setup

Throughout this post, we will use the Causal Scrubbing setup (with notation similar to Geiger et al. (2023)):

^{[1]}Π describes which node in H is supposed to correspond to which node in L. We'll write inputs to Π as subscripts, ΠX:=Π(X) for a node X∈H. H typically needs to have the same input and output types as L (with causal scrubbing being an exception, as we'll see).In this setting, all three methods define what it means for (H,Π) to be a correct explanation, as well as graded notions that can quantify “how incorrect” H is. (We will often just refer to the explanation as H, even though Π is also part of it.)

Some notes on this:

setof nodes in L). They also don't require Π to be a homomorphism. Locally consistent abstractions were originally formulated even more generally, for computations other than static graphs (e.g. Turing machines). But even the simple setting we use in this post leads to many interesting differences between methods.aspectof the output (i.e. up to an equivalence relation), we make that choice part of L by adding a final node at the end that “forgets” all the information we don’t want to explain. For example, causal scrubbing is usually applied to explanations of thelossinstead of the full output of the network. Similarly, if the explanation H only takes in part of the input, we implement that as part of H, by just ignoring the other parts.## Overview of different methods

This section will explain all the methods we're comparing and already highlight many similarities and differences. We recommend at least skimming it even if you're already familiar with all of these methods in their own right.

## Causal abstractions

Work on abstracting causal models started without any connections to explaining neural networks. None of the concrete algorithms we'll compare exactly implements these causal abstractions. But they can serve as an overarching framework (and in the case of the Geiger et al. line of work, they are the explicit motivation for the algorithm).

Why would causality be relevant for discussing computations like neural networks? The key observation is that acyclic deterministic causal models are the same thing as static computational graphs, and interventions in these causal models correspond to ablations in computational graphs. The difference is only one of terminology and framing.

The key idea of causal abstractions is that we can either do an intervention in the low-level model L and then translate the result to the high-level model H, or first translate the intervention to H and then perform it there. If H is a good causal abstraction, these two should be equivalent.

There are several somewhat different formalizations of this idea. We follow Geiger et al. (2023) in describing a simplified version of

constructive abstraction(Beckers & Halpern, 2019) specifically for deterministic models:- In addition to H and Π:H→L, we also need maps τX:Val(ΠX)→Val(X) for every high-level node X∈H, which map low-level values to high-level values. We write τ:Val(L)→Val(H) for the combination of all these maps. On output nodes, τ is constrained to be the identity.
- The computational graphs L and H induce functions that take their inputs to the joint setting of all their nodes (simply by doing a forward pass). We'll call this function FL:Inputs(L)→Val(L) and FH analogously.
- For any (hard)
for L - (H,Π,τ) is a valid causal abstraction of L if for all interventions i, we have

FHτ(i)=τ∘FLi.^{[2]}interventioni,i.e. a setting of some of the nodes, we can define the intervened causal graph Li the usual way. We write τ(i) for the corresponding intervention in H. If i sets a node Z∈L to value z and Z is in the image of Π, then τ(i) sets Π−1(Z) to τ(z). If Z is not in the image of Π, it is ignored for τ(i).We can (somewhat informally) write this as a commutative diagram:

For readers who have seen constructive abstractions described elsewhere, we note that our version might look different because of several simplifying assumptions: we only deal with deterministic acyclic causal graphs, and unlike Geiger et al. (2023) we assume that every node in H corresponds to only one node in L. This means that

everyintervention i in L has some analogous intervention τ(i) in H (whereas in the more general version, you need to worry about interventions that do different things to several nodes in L that all correspond to a single node in H).## Interchange interventions/Resampling ablations

As we just saw, a causal abstraction consists not only of H and Π, but also of τ-maps. In the context of neural network interpretability, these would map neural representations to interpretable values in H. Specifying these explicitly would be a lot of work—we may instead want to test hypotheses of the form "this neural representation encodes [meaningful feature]" without making claims about

howthat feature is encoded.Interchange interventions(Geiger at al., 2021a) let us test hypotheses like that usingimplicitlydefined τ-maps. The idea is that we can generate interventions in L by replacing the activations of some nodes with those on different inputs. Then we get the corresponding intervention in H just by using the same inputs to replace activations in the corresponding nodes.This is exactly the same idea that's called

resampling ablationsin the context of causal scrubbing. The full causal scrubbing algorithm has additional components, but it can be understood as using interchange interventions—more on that below.In the setting for this post, an interchange intervention works as follows:

base inputb andsource inputss1,…,sk from D.This procedure lets us generate equivalent interventions on L and H without needing to explicitly compute τ-maps. It's not a complete algorithm though, as we will also need to check whether the

effectsof these interventions are what the explanation predicts.## The Geiger et al. algorithm

How can we check whether the causal abstraction condition is satisfied using interchange interventions? The answer is that in practice, we don't: we only check a weaker condition. But first, let's understand the key difficulty.

Interchange interventions let us generate matching interventions in both graphs, i.e. pairs (i,τ(i)), without explicitly representing or using τ. But if we look at the commutative diagram from above, τ is also used on the right side to compare the effects of the interventions in the two graphs. It's much less clear how to do that efficiently without representing τ:

The difference here is that on the left side, we only need to be able to

generatelots of matching interventions. But then for each of these interventions, we get some specific values in L and H, and we need todecidewhether these values match.So what Geiger et al. do instead is to only compare the

outputsof L and H after the interchange intervention. Since we assumed these live in the same space, this is much easier and doesn't require τ-maps.Concretely,

Interchange Intervention Accuracy(IIA) (Geiger et al., 2021b) is the probability that the outputs are exactly equal on a random interchange intervention. Of course other metrics are possible, e.g. an expected distance between outputs (as Geiger at al. (2023) point out, IIA is a special case of that with a 0-1-distance).Because we only compare outputs, an IIA of 100% does

notimply that H is a valid causal abstraction.^{[3]}We will discuss this more and give an example later.## Causal scrubbing

Causal scrubbing also uses interchange interventions (or "resampling ablations"), but the full algorithm differs from the Geiger at al. one in two key ways.

First, causal scrubbing

only intervenes inL, not in H. So how can we check whether the intervention did what H predicts, without actually intervening on H? The answer is simple: weonly perform interventions that leaveHunchanged. The causal scrubbing algorithm recursively uses rejection sampling to construct interchange interventions that leave all the values in H unchanged. If we perform the corresponding interventions in L, that may very well change intermediate activations in L. But, if the explanation is correct, it should not change theoutputof L.So just like the Geiger et al. algorithm, causal scrubbing compares only outputs (as opposed to intermediate values). But because of the restricted space of interventions it performs, it can simply compare the output of L on a base input to the new output after an interchange intervention, instead of comparing to the output of H. The only role of H in causal scrubbing is to determine

which interventions to perform. Because of that, it's fine for the output type of H to be different than that of L in the case of causal scrubbing. For example, L might output a continuous loss, whereas H only outputs discrete values.In terms of what we've discussed so far, causal scrubbing would simply perform strictly fewer interventions, and would be a weaker test. However, causal scrubbing has a second key component:

treeificationof the computational graph.To understand treeification and why it matters, we'll visualize interchange interventions by mixing colors as in the following figure:

The left figure visualizes a normal forward pass on a single blue input. On the right, we have done an interchange intervention on B that patches in the activation from a red source input. The resulting output D is a mix of both colors, it depends on the blue and the red input.

Now let's look at what happens as we intervene on more nodes simultaneously:

Here, we have performed an interchange intervention on both B and C, using two different source inputs (red and green). The key observation is that the output D now doesn't depend on the blue base input anymore. B and C completely screen off the effects of A on D. In fact, there is no intervention we could perform on this graph where the output would be affected by a mix of three different inputs.

Treeification lets us mix more inputs than a computational graph would ordinarily allow. The idea is that the treeified graph should have the same

pathsfrom input to output as the original graph, but separated into a tree:Our graph has three paths from input to output: A-B-D, A-B-C-D, and A-C-D. As you can see, the treeified version has the same paths, but we've made copies of all the nodes as necessary. For example, B occurs in two of the paths, so there are two copies, B1 and B2.

The figure above shows how to do the last interchange intervention we saw on the treeified graph. But because of the treeification we can also do an additional interchange intervention that mixes three different inputs:

Note that on treeified graphs, we can get any interchange interventions by just computing a forward pass on mixed inputs. We don't actually need to intervene on intermediate nodes: every node has a set of corresponding inputs, and we can just change those inputs without affecting any other downstream nodes. We'll still call this an interchange intervention since it's exactly equivalent in terms of the effects.

Combining what we've discussed so far, causal scrubbing works roughly as follows:

There are two subtleties that make this description imprecise:

averageboth the unchanged and scrubbed outputs over all base inputs and then compare these expectations. In other words, we compute ∣∣E[Lscrubbed(b,si)]−E[L(b)]∣∣ instead of E∣∣Lscrubbed(b,si)−L(b)∣∣. Again see here for a brief discussion on this choice. We will also analyze this more in the next section.In practice, causal scrubbing can be implemented by a recursive algorithm, as opposed to explicitly constructing the treeification. The resulting recursive call graph corresponds one-to-one to the treeified L.

## Locally consistent abstractions

Both the Geiger et al. approach and causal scrubbing use interchange interventions to avoid explicitly representing τ-maps. In contrast, locally consistent abstractions (LCAs)

dorepresent τ-maps explicitly. In practice, τ-maps will likely not be specified as part of an interpretability hypothesis. Instead, they can belearned: if there is any τ-map for which an explanation H would be valid, then we accept H. The only constraint is that, as for causal abstractions, the τ-map on output nodes must be the identity.The other key difference is that LCAs only check whether the

immediateeffects of an intervention are the same between low-level model and high-level model. In this sense, they can be seen as simplifying causal abstraction in another direction than Geiger et al. or causal scrubbing do:The following figure contrasts those approaches, the nodes for which consistency is checked are circled in yellow:

Let's describe LCAs more formally in their own right. For any node X, we'll write fX for the function that computes the value of X from its parents. Then (H,c,τ) is a locally consistent abstraction if for every node X in H, we have

fX∘τ=τ∘fΠX.We are abusing notation slightly here, see footnote.

^{[4]}We can write this as a commutative diagram again:pa(X) are the parents of node X and Val(X) the set of possible values X can take, analogously for the other cases.

Intuitively, this is saying we can either perform a computational step in the low-level model (fΠX) and then abstract with τ, or we can first abstract and then perform a computational step in the high-level model (fX). For good abstractions, these should be equivalent.

We haven't explicitly mentioned interventions here. Simply computing fX is of course equivalent to intervening on pa(X) and then evaluating X, but the language of causality and interventions isn't really necessary if we only want to describe

localconsistency.So far, we've simply stated the consistency condition as equality of two functions. But realistically, we often can't check exact equality, we can only test the consistency condition on samples (like in the earlier methods). There are two options for how to get activations for pa(ΠX):

(Of course, yet other options exist, e.g. computing all "unimportant" parents from a single input but all important parents from different ones, similar to causal scrubbing.)

Finally, note that if the consistency condition

wassatisfied exactly and on all possible inputs, then this would also imply global consistency in the causal abstraction sense. But with restricted activation distributions, and with approximate consistency, local and global methods differ in several aspects. We'll discuss this more in the next section.## Axes of differences between methods

Hopefully, you now have a sense of the differences between causal abstraction, the specific implementation by Geiger et al., causal scrubbing, and LCAs. In this section, we'll identify several "themes" in how these methods differ, and discuss the consequences of those differences.

## Set of accepted explanations

Throughout this discussion, we'll be asking: "How strict is this notion of correctness, i.e. which explanations does it accept?" Here are the results upfront:

It's not obvious that we always want the strictest possible definition—we'll briefly discuss this question later.

## Checking consistency for intermediate nodes vs output only

Causal abstraction and LCA check consistency conditions for intermediate nodes of the computational graphs, whereas the concrete Geiger et al. algorithm and causal scrubbing only look at the output.

Looking only at the output means accepting some weird explanations that causal abstractions and LCA would reject. For example, say that the output node of both L and H is simply constant. In that case, H can have arbitrary circuits before the output and will be accepted as a correct explanation. A constant output is an extreme case, but similar things can happen much more generally: H can have "hallucinated circuits" as long as they don't actually impact the final output.

Having mechanisms in H that don't affect the output is a strange property, and perhaps we could rule out such explanations a priori (e.g. by favoring simpler explanations). We don't know whether such an approach could fully bridge the gap between output-based methods and causal abstraction.

## Explicit vs implicit τ-maps

Whether to represent τ-maps explicitly is mostly a choice of implementation that affects what you can efficiently do. The explicit τ-maps are what allow LCAs to compare intermediate nodes, as we just discussed. On the other hand, there is of course some overhead from learning these maps, and the practical difficulty of optimizing them may reject extra hypotheses.

## Global vs local consistency and activation distributions

All methods except LCA check the effects of interventions on nodes that are arbitrarily far downstream, i.e. they check

globalconsistency. One of the effects of that is that they check consistency on activations that are "off-distribution", i.e. not induced by forward passes on the input distribution D.Off-distribution activations arise in a two-step process:

different inputsx1 and x2 (because of an interchange intervention). At this point, theoverall stateof the computation might already be off-distribution: normally, n1 and n2 would both be computed from the same input, and computing them from different inputs might have destroyed some mutual information between them. But the activation of individual nodes is still on-distribution.We can get the step 1. off-distribution activations quite easily for LCA by sampling activations for parents in L using separate inputs. Step 2. off-distribution activations might then occur in the node X for which we're checking consistency, but crucially, these activations won't be fed into any other mechanisms fY, only into τ. So we're only testing the causal mechanisms fX on-distribution (at least in terms of the marginal distribution for each input to fX).

This leads to LCA accepting some explanations that all the other methods reject. A typical example is a node X in L that has the same value v1 on all inputs from D, but a different value v2 on other inputs. LCA will accept certain explanations that claim X is

mechanisticallyconstant (as opposed to being computed by some complicated mechanism that happens to always give the same result). The other methods will sometimes reject such an explanation, depending on the actual implementation of X.A related example are duplicate mechanisms in L: if L has two copies of the same circuit and then just takes the average at the end, LCA will often accept an explanation that only points out one of the circuits, while the other methods will reject that explanation.

## Only allowing interventions that leave H unchanged

Recall that causal scrubbing only allows interventions that don't change any of the values in the explanation H. This leads to examples where causal scrubbing accepts an explanation that all the other methods reject. Consider the following type of explanation H:

Effectively, H doesn’t do anything, it just keeps the input around indefinitely. But causal scrubbing always accepts this “explanation”: since H claims that all the nodes matter, and that all the information in every node matters, it doesn’t allow any ablations at all.

## Expectations vs sample-wise errors

Causal scrubbing compares the

expectationsof the scrubbed output and the original one, whereas all the other methods compute errors for each base input and then take the expectation over these errors. The second approach is stricter in terms of which explanations are accepted. With the causal scrubbing metric, errors can "cancel", see here for an example.## Treeification

Treeification is the one way in which causal scrubbing is stricter than all the other methods. Recall the figure above where we showed how treeification allows "mixing" more different inputs than is possible using only interventions on the original graph. In some cases, this should mean causal scrubbing will test on activations that causal abstractions doesn't generate, so it may reject explanations that fail on these activations.

## Computational complexity

Because of the (implicit) treeification, computing a single scrubbed output for causal scrubbing can take time exponential in the size of the explanation H. In contrast, LCA and interchange interventions on non-treeified graphs take linear time.

One subtlety is that there are exponentially many different interventions one could sample in the Geiger et al. approach. In practice, sampling only a reasonably small number of interventions plausibly seems good enough, but we're not entirely sure to what extent that retains benefits of doing global interventions.

One could also use approximations of causal scrubbing that only trace out a small random part of the treeification of L, to avoid exponential time complexity. Again, we don't know whether this would preserve the higher strictness that treeification usually gives.

While LCAs are linear time in the size of L and H, there are two caveats:

learningthe τ-maps will add some overhead. This overhead might be reasonably small if e.g. linear τ-maps suffice and the explanation is significantly smaller than the full network.## Compounding errors and chaotic computations

The examples so far showed discrepancies between methods in terms of when they judge an explanation as

perfectlycorrect. There is one additional qualitative difference once we consider graded judgments ofhow incorrectan explanation is. Namely, LCAs computelocalerrors, which don’t compound throughout the computation. To get a single metric, we justadd upall the local errors. For all the other methods, if an intervention early on in the computation has slightly different effects than predicted by H, these effects can compound to get a very large error.This difference becomes very pronounced if L implements a chaotic computation. For example, say that L simulates a double pendulum using 64-bit floating point numbers, and H implements the exact same computation using 32-bit floating point numbers. Under LCAs, this explanation has only a rather small error. In contrast, the explanation is completely wrong according to causal scrubbing or interchange intervention accuracy, assuming the simulation runs for long enough that 32-bit errors compound a lot.

## Multiple simultaneous interventions

Earlier versions of interchange interventionsallowed interventions encompassing only a single abstract node at a time. This leads to accepting some incorrect hypotheses that are rejected by causal scrubbing and bylater versions of interchange interventions, see the appendix for an example. The problems don’t seem specific to interchange interventions, and multiple simultaneous interventions seem desirable for any similar method. In their latest versions, all methods reject these types of examples.## Discussion

"I" in this section refers to Erik and these are specifically my personal takes.## A common framework

Causal abstractions seem like a promising way of unifying all these approaches. LCAs and the Geiger et al. approach can both be seen as weakening the causal abstraction condition in different ways. And causal scrubbing is still very similar except for working on the treeified graphs.

On the other hand, I do think there's room for several

differentframeworks or perspectives here. For example, LCAs don't actually have a very causal feel, but seem very natural to me from other perspectives.## Combining components in different ways

My sense is that the specific algorithms people have come up with are largely historical accidents, and you could just as well combine some of the components in different ways. For example, the choice of comparing expectation or sample-wise errors has nothing to do with whether you want to do treeification.

It's not entirely arbitrary that the purely local method (LCA) uses explicit τ-maps and the others don't (since local consistency checks require τ-maps, whereas if you check global effects of interventions, you can just look at the output). But I think you could totally have

bothexplicit τ-maps and global consistency checks, if that's the level of strictness you want (and you're willing to pay a bit of computational overhead). That would exactly correspond to causal abstraction.## How strict do we want our criterion to be?

Given that the different methods all seem incomparable, we could get a particularly strict criterion by only accepting if all of them accept (or by directly doing full-blown causal abstraction on treeified graphs).

It's not clear to me that we should though:

wantyour explanation to fail on some out-of-distribution inputs. Being too strict could mean you force the explanation to contain the anomalous mechanism you want to detect (in which case detection won't work).## Different conceptual goals

I think of the global approaches (causal scrubbing/interchange interventions) as trying to only accept explanations that contain

allthe mechanisms relevant for the input-output behavior. So for example, if there are two copies of a mechanism, both need to be included in the explanation. On the flip side, causal scrubbing and the Geiger et al. approach, because they only look at the output, will accept some explanations that "hallucinate" additional mechanisms.LCAs accept an explanation as long as it is

sufficientto explain the output and containsat mostthe mechanisms that are actually present. So if there are two exact copies of a mechanism (that e.g. get averaged at the end), it will accept an explanation that only contains one of them. But if the explanation contains hallucinated mechanisms, it will be rejected.Full causal abstraction combines both of these, accepting an explanation only if it contains exactly the mechanisms that are important for the output (and potentially irrelevant ones, but not hallucinated ones). A big caveat though is that I'm pretty confused about how "important for the output" should be interpreted if we only want to explain behavior on some distribution D. As long as we aren't literally interested in interventions (i.e. only care about behavior on actual inputs), it's not obvious that e.g. a second copy of a mechanism is "important for the output".

## Conclusion

On the one hand, we have seen that causal scrubbing, causal abstractions, and locally consistent abstractions are more similar than they may appear based on their original expositions. In practice, it seems likely they will often give similar results.

On the other hand, there are differences between all of these methods, and each one of them accepts some hypotheses that are rejected by one of the other methods. We could combine all of them to get one stricter definition of "correct explanations". However, the differences between the approaches highlight conceptual questions about what it means to explain a behavior "on a distribution", so it's not obvious that getting the strictest definition we can is desirable.

## Appendix

Early papers in the Geiger et al. line of work only allowed for one abstract node to be intervened on at a time, which makes the method less strict.

For instance, consider an example where we have numerous copies of the same computation stacked together in a graph L, while the output of L is a value produced by the majority of copies (which will always be the same without interventions). We can come up with abstractions for each of these duplicates that are individually performing distinct tasks, but together perform a computation that is a seemingly perfect causal abstraction of L, if we’re only allowed to perform one intervention at a time.

For a concrete example, consider the following figure:

Here for any base input b and source input s and any node X∈L to be intervened on, the output of L after the intervention is I[5≤b≤10]. We'll show this is also true for H.

If the intervention is on a yellow node, H outputs

{True,I[s≥5]+I[b≥5]+I[b≤10]+I[b≤10]+I[5≤b≤10]≥4False,otherwiseIf 5≤b≤10, then this expression is true. If b<5, then the 2nd and the 5th indicators are false and the whole expression is false. If b>10, then the 3rd and 4th indicators are false. So the output of H matches L for interventions on one of the yellow nodes, and analogously for the red nodes.

It's also obvious that the output of H on a blue intervention is I[5≤b≤10] since if 5≤b≤10, then the first four indicators are true, otherwise at least 2 of them are false.

Therefore, this abstraction would be accepted by doing interchange interventions that intervene on only one node at a time. On the other hand, causal scrubbing or causal abstractions with multiple interventions don’t accept this abstraction (as long as the input distribution isn't too impoverished).

^{[5]}We came up with this example while figuring out to what extent multiple interventions are necessary. Turns out even in a very simple setting with 5 nodes, methods that allow only one intervention might fail. We therefore think it is necessary for all further intervention methods to permit multiple simultaneous interventions.

^{^}Some versions of causal scrubbing allow the correspondence Π to map vertices u,v∈H with an edge u→v to non-adjacent Πu,Πv, so long as a

pathfrom Πu to Πv exists; which is technically not a graph homomorphism. This post follows the published setup for causal scrubbing, and restricts Π to be a graph homomorphism.^{^}Recall that we assume H and L have the same output type, so the identity function makes sense. If we didn't constrain the τ-maps on outputs, an explanation could choose a constant τ-map for the output, and thus wouldn't actually need to explain anything. As mentioned in the Setup section, if we only want H to explain some aspect of the output, we make that choice a part of L.

^{^}Note that Geiger et al., (2021b) have a result showing that 100% IIA

doesimply a valid causal abstraction. However, this is for a high-level model where all but one intermediate node have been marginalized. We are saying that the full high-level model H with multiple intermediate nodes need not be a causal abstraction.^{^}The two copies of τ are really defined over different domains, and there should technically be a projection map that discards the values of any "unimportant" parents, i.e. those that don't have corresponding nodes in H. So the more pedantic version would be

fX∘τpa(X)∘πΠ(pa(X))=τX∘fΠX,where pa(X) is the set of parents of X.

^{^}Consider, for instance, causal scrubbing with a uniform distribution D over {1,7,18}. Note this example would work with other input spaces that have at least one value below 5, above 10 and in the interval [5, 10] each, but for simplicity we consider the minimal possible input space.

It is clear that Ex∼D[L(x)]=13. At the same time, the scrubbed expectation would equal E[Lscrubbed]=1148≠13: if b=1, then the value of the yellow nodes is False. There is only one value in D that agrees with b on yellow nodes, which is 1. Thus, the value of the first two nodes in G shall be false. Similarly, the value of the 5th node is also False and Lscrubbed(b,s)=0. The same argument works if b=18. If b=7, then both 7 and 18 agree on yellow nodes, so the value of the first two nodes in G is equally probably either True or False. Same for the 3rd and 4th nodes. The 5th node is always True. Since the output of G is the majority of nodes, it will output True in (42)+(43)+(44) = 11 cases. Thus, the scrubbed expectation equals Escrubbed=13⋅0+13⋅0+13⋅1116=1148.