Causal Abstraction Inference under Lossy Representations

The study of causal abstractions bridges two integral components of human intelligence: the ability to determine cause and effect, and the ability to interpret complex patterns into abstract concepts. Formally, causal abstraction frameworks define connections between complicated low-level causal models and simple high-level ones. One major limitation of most existing definitions is that they are not well-defined when considering lossy abstraction functions in which multiple low-level interventions can have different effects while mapping to the same high-level intervention (an assumption called the abstract invariance condition). In this paper, we introduce a new type of abstractions called projected abstractions that generalize existing definitions to accommodate lossy representations. We show how to construct a projected abstraction from the low-level model and how it translates equivalent observational, interventional, and counterfactual causal queries from low to high-level. Given that the true model is rarely available in practice we prove a new graphical criteria for identifying and estimating high-level causal queries from limited low-level data. Finally, we experimentally show the effectiveness of projected abstraction models in high-dimensional image settings.

Determining cause and effect is important since it helps us understand the difference between whether eating vegetables prevents cancer or whether individuals who happened to eat vegetables were less likely to smoke, therefore preventing cancer. Determining this from data is difficult, so it may be helpful to work more abstractly: for example, choosing to think about calories rather than individual nutrients like carbohydrates, fat, and protein. The problem is that sometimes important details may be abstracted away, like perhaps the amount of fat is relevant for determining cancer risk.Our paper develops a generalized approach to abstractions that accommodates this loss of information by treating the abstract quantities as distributions over the low-level details they are abstracting. For example, in an image of a dog, one can see many details of the dog, like the breed, the texture and color of its fur, and its posture. If the pixels were simply abstracted away as a single word label "dog", we lose all of this information. There may be many dog images in the dataset, each with a different breed. Still, if breed is important information, our paper allows one to solve two problems: (1) any causal calculations performed on this dataset can still be done by simply randomly sampling over different dog breeds with probabilities based on which breeds are more common, and (2) despite the lack of details, the image can still be reconstructed by simply sampling whatever is lost in the abstraction, allowing for a full high-quality dog image from simply the label "dog".