Reference Compositional abstraction error and a category of causal models [rischel_compositional_2021]

Abstract

Interventional causal models describe several joint distributions over some variables used to describe a system, one for each intervention setting. They provide a formal recipe for how to move between the different joint distributions and make predictions about the variables upon intervening on the system. Yet, it is difficult to formalise how we may change the underlying variables used to describe the system, say moving from fine-grained to coarse-grained variables. Here, we argue that compositionality is a desideratum for such model transformations and the associated errors: When abstracting a reference model M iteratively, first obtaining M' and then further simplifying that to obtain M'', we expect the composite transformation from M to M'' to exist and its error to be bounded by the errors incurred by each individual transformation step. Category theory, the study of mathematical objects via compositional transformations between them, offers a natural language to develop our framework for model transformations and abstractions. We introduce a category of finite interventional causal models and, leveraging theory of enriched categories, prove the desired compositionality properties for our framework. 

@misc{rischel_compositional_2021,
 author = {Rischel, Eigil F. and Weichwald, Sebastian},
 date = {2021-08-05},
 doi = {10.48550/arXiv.2103.15758},
 eprint = {2103.15758 [cs, math, stat]},
 eprinttype = {arxiv},
 keywords = {Computer Science - Artificial Intelligence, Computer Science - Logic in Computer Science, Computer Science - Machine Learning, Mathematics - Category Theory, Statistics - Machine Learning},
 number = {{arXiv}:2103.15758},
 publisher = {{arXiv}},
 title = {Compositional Abstraction Error and a Category of Causal Models},
 url = {http://arxiv.org/abs/2103.15758},
 urldate = {2024-05-25}
}

Backlinks

Bidirectionality in graphical models [efr-0002]

Definition Structural Causal Model [efr-0003]

Let G be a (finite) directed acyclic graph (DAG). Then a structural causal model (SCM) on G consists of

  • For each vertex v \in V(G), measurable spaces \mathcal {X}_v, \mathcal {E}_v
  • A family of independent random variables E_v \in \mathcal {E}_v---these are called the exogenous variables
  • For each v, a measurable function \phi _v: \mathcal {E}_v \times \prod _{v' \to v} \mathcal {X}_{v'} \to \mathcal {X}_v (note that if v has no parents, the product is just a singleton)

There is a significant existing literature (see Abstracting causal models, Reference [rischel_compositional_2021], Reference [rubenstein_causal_2017]. For a survey see Reference [zennaro_abstraction_2022]) studying transformations between structural causal models. These have generally been called something like abstractions, the typical examples being the relationship between a high-level and a low-level model of the same system. However, the precise properties that we should ask for in such a transformation have turned out to be somewhat subtle. The most obvious condition to impose is to ask, for any possible variable we could intervene on, say X, and any other variable (or collection of variables) Y, that the diagram

of kernels commutes---in other words, given an intervention in the low-level model, the two possible distributions in the high-level model agree.

The main problem with this condition is that it is simply too strict. We generally can't ask that every low-level intervention is well-modeled by the corresponding high-level intervention. Rather, each high-level intervention corresponds to some distribution of the corresponding low-level variables, and we must ask that a diagram of this form:

commutes. For example, if the low-level variables are the velocities of all the gas molecules in a container, and the high-level variables are thermodynamic quantities like temperature and pressure, we can't expect that every set of velocities corresponding to a given temperature will lead to a given distribution of outcomes---rather, intervening on the temperature leads to some particular distribution of possible sets of velocities, and this leads to the observed distribution of outcomes.

The key here is that control information flows in the opposite direction from measurement information. This phenomenon repeats in many examples.

List of notions of "model abstraction" [lcc-0015]

The morphisms in Categorical Systems Theory

There are actually two notions of morphism between systems, here. In both cases, you have a mapping on the "state space", and some sort of mapping on the input/output interface which needs to be compatible.

In the case of lenses, you have a map forwards on the output, and a map backwards on the input - this clearly allows you to change the interface of the system, and then you require that the map on state is dynamics-preserving after changing interface.

In the case of charts, everything goes in the same direction, and you can ask for compatibility between the systems, but you don't have a change-of-interface functor.

Abstractions of causal DAG models

See eg Visual causal feature learning, Multi-level cause-effect systems, Causal feature learning: An overview, Approximate causal abstraction, Compositional abstraction error and a category of causal models.

There is a complicated literature on these, but generally they are maps between the outcome space of causal models which in some sense preserve the interventional distributions.

There is a question of bidirectionality, discussed through these papers (eg in Compositional abstraction error and a category of causal models), which precisely mirrors the distinction between lenses and charts in Categorical Systems Theory. Namely, in order to interpret high-level interventions as low-level interventions, we need a map "backwards" (this is muddled by the fact that the set of inputs, possible interventions, is usually equal to the set of outputs in this case).

Abstractions of Kripke Modal Transition Systems

A KMTS is a "partially-defined" transition system, and we are interested in binary relations from one such to another which have the property of being "directed bisimulation" with respect to the partiality - i.e, morally speaking, it would be a bisimulation if we replaced one system with a "more specific" one. See eg Automatic Abstraction Using Generalized Model Checking