Author's Notes: This post is fairly technical, with little background and minimal examples; it is not recommended for general consumption. A general understanding of causal models is assumed. This post is probably most useful when read alongside the paper. If your last name is "Beckers" or "Halpern", you might want to skip to the last section.

There’s been a handful of papers in the last few years on abstracting causal models. Beckers and Halpern (B&H) wrote an entire paper on definitions of abstraction on causal models. This post will outline the general framework in which these definitions live, discuss the main two definitions which B&H favor, and wrap up with some discussion of a conjecture from the paper. I'll generally use notation and explanations which I find intuitive, rather than matching the paper on everything.

In general, we’ll follow B&H in progressing from more general to more specific definitions.

General Framework

We have two causal models: one “low-level”, and one “high-level”. There’s a few choices about what sort of “causal model” to use here; the main options are:

• Structural equations
• Structural equations with a DAG structure (i.e. no feedback loops)
• Bayes nets

B&H use the first, presumably because it is the most general. That means that everything here will also apply to the latter two options.

Notation for the causal models:

• We’ll write $X^H$ for the variables in the high-level model and $X^L$ for variables in the low-level model.
• We’ll use capital-letter indices to indicate choosing multiple indices at once. For instance, with $S=$ $(1,2,3)$, $X_S$ would be $(X_1,X_2,X_3)$.
• We’ll write interventions as $X_S\leftarrow X_S^{\ast }$. For instance, with $S=(1,2,3)$, $X_S\leftarrow X_S^{\ast }$ would be equivalent to the three simultaneous interventions $(X_1\leftarrow X_1^{\ast },X_2\leftarrow X_2^{\ast },X_3\leftarrow X_3^{\ast })$. Usually both $S$ and $X^{\ast }$ will be unspecified, to indicate a generic intervention.

Next, we need some connection between the high-level and low-level model, to capture the intuitive notion of “abstraction”. At its most general, this connection has two pieces:

• A mapping $\tau$ between values of the variables in the models: $X^H=\tau \left(X^L\right)$
• A mapping $\omega$ between interventions: $(X_{S^H}^H\leftarrow X_{S^H}^{H\ast })=\omega (X_{S^L}^L\leftarrow X_{S^L}^{L\ast })$. Here $\omega$ determines both $S^H$ and $X_{S^H}^{H\ast }$ as a function of $S^L$ and $X_{S^L}^{L\ast }$.

Note that, for true maximum generality, both $\tau$ and $\omega$ could be nondeterministic. However, we’ll generally ignore that possibility within the context of this post.

Finally, the key piece: the high-level and low-level models should yield the same predictions (in cases where they both make a prediction). Formally:

$P\left[X^H|do\right(\omega (X_S^L\leftarrow X_S^{L\ast })\left)\right]=P\left[\tau \right(X^L\left)|do\right(X_S^L\leftarrow X_S^{L\ast }\left)\right]$

For the category theorists: this means that we get the same distribution by either (a) performing an intervention on the low-level model and then applying $\tau$ to $X^L$, or (b) first applying $\tau$ to $X^L$, then applying the high-level intervention (found by transforming the low-level intervention via $\omega$).

The first definition of “abstraction” examined by B&H is basically just this, plus a little wiggle room: they don’t require all possible interventions to be supported, and instead include in the definition a set of supported interventions. This definition isn’t specific to B&H - it’s an obvious starting point for defining abstraction on causal models as broadly as possible. B&H adopt this maximally-general definition from Rubenstein et al, and dub it “exact transformation”.

B&H then go on to argue that this definition is too general for most purposes. I won’t re-hash their arguments and examples here; the examples in the paper are pretty readable if you’re interested. They also introduce one slightly stronger definition which I will skip altogether; it seems to just be cleaning up a few weird cases, without any major conceptual additions.

τ-Abstraction

The main attraction in B&H is their definition of “$\tau$-abstraction”. The main idea in jumping from the maximally-general framework above to $\tau$-abstraction is that the function $\tau$ mapping low-level variables to high-level variables induces a choice of mapping between interventions; there’s no need to leave the choice of $\omega$ completely open-ended.

In particular, since $X^H=\tau \left(X^L\right)$ by definition, it seems like $\tau$ should also somehow relate $X^{H\ast }$ to $X^{L\ast }$ in the interventions $X_{S^L}^L\leftarrow X_{S^L}^{L\ast }$ and $X_{S^H}^H\leftarrow X_{S^H}^{H\ast }$. The obvious condition is $X^{H\ast }=\tau \left(X^{L\ast }\right)$. However, the interventions themselves only constrain $X^{H\ast }$ and $X^{L\ast }$ at the indices $S^H$ and $S^L$ respectively, whereas $\tau$ may depend on (and determine) the variables at other indices.

One natural condition to impose: each value of $X^{H\ast }$ consistent with the high-level intervention should correspond to at least one possible value of $X^{L\ast }$ consistent with the corresponding low-level intervention, and each possible value of $X^{L\ast }$ consistent with the low-level intervention should produce a value of $X^{H\ast }$ consistent with the high-level intervention. More formally: if our intervention values are $X_{S^H}^{H\ast }=x^{H\ast }$ and $X_{S^L}^{L\ast }=x^{L\ast }$, then we want equality between sets:

$\{X^{H\ast }|X_{S^H}^{H\ast }=x^{H\ast }\}=\left\{\tau \right(X^{L\ast })|X_{S^L}^{L\ast }=x^{L\ast }\}$

This is the main criterion B&H use to define the “natural” mapping between interventions ${\omega }_{\tau }$. (The exact definition given by B&H is a bit dense, so I won’t walk through the whole thing here.)

Armed with a natural transformation ${\omega }_{\tau }$ between low-level and high-level interventions, the next step is of course to define a notion of abstraction: modulo some relatively minor technical conditions, a $\tau$-abstraction is an abstraction consistent with our general framework, and for which $\omega ={\omega }_{\tau }$.

One more natural step: A “strong” $\tau$-abstraction is one for which all interventions on the high-level model are allowed.

Constructive τ-Abstraction

In practical examples of abstraction, the high-level variables $X^H$ usually don’t all depend on all the low-level variables $X^L$. Usually, the individual high-level variables $X_i^H$ can each be calculated from non-overlapping subsets of the variables $X^L$. In other words: we can choose a partition $\sigma$ of the low-level variables and break up $\tau$ such that

$X_i^H={\tau }_i\left(X_{{\sigma }_i}^L\right)$.

Also including all the conditions required for a strong $\tau$-abstraction, B&H call this a “constructive” $\tau$-abstraction.

The interesting part: B&H conjecture that, modulo some as-yet-unknown minor technical conditions, any strong $\tau$-abstraction is constructive.

I think this conjecture is probably wrong. Main problem: constructive $\tau$-abstraction doesn’t handle ontology shifts.

My go-to example of causal abstraction with an ontology shift is a fluid model (e.g. Navier Stokes) as an abstraction of a particle model with only local interactions (e.g. lots of billiard balls). In this case, we have two representations of the low-level system:

• A Lagrangian representation, in which we track the position and momentum of each particle
• An Eulerian representation, in which we track the mass and momentum densities as a function of position

The two are completely equivalent; each contains the same information. Yet they have very different structure:

• In the Lagrangian representation, each “variable” (i.e. a particle’s mass & momentum at a given time) interacts with all other variables which are nearby in time; we need to check for collisions against every other particle, even those far away in space, since we don’t know ahead of time which will be close by.
• In the Eulerian representation, each “variable” (i.e. mass & momentum density at a given point in space and time) interacts only with variables which are nearby in both space and time.

In this case, the high-level fluid model is a constructive abstraction of the Eulerian representation, but not of the Lagrangian representation: the high-level model only contains interactions which are local in both time and space.

Conceptually, the problem here is that our graph can have dynamic structure: the values of the variables themselves can determine which other variables they interact with. When that happens, an ontology shift can sometimes make the dynamic structure static, as in the Lagrangian -> Eulerian transformation. But that means that a constructive $\tau$-abstraction on the static structure will not be a constructive $\tau$-abstraction on the dynamic structure (since the partition would depend on the variables themselves), even though the two models are equivalent (and therefore presumably both are $\tau$-abstractions).

This does leave open the possibility of weakening the definition of a constructive $\tau$-abstraction to allow the partition $\sigma$ to depend on $X^L$. Off the top of my head, I don’t know of a counterexample to the conjecture with that modification made.