annotated-sfm

Structural Functional Models (SFM) for defining causality.

Causality is just functions: "x causes y" roughly means y=f(x, ...).

Paper: Reducing Causality to Functions with Structural Models (https://arxiv.org/abs/2307.07524)

This is a simple toy algorithm that illustrates the philosophical concept of "defining causality as functions." It includes algorithms that I didn't fully specify in the paper.

For practical causal inference in machine learning, please refer to specialized libraries like DoWhy and CausalML. I personally recommend Elements of Causal Inference: Foundations and Learning Algorithms for a good introduction book.

This repository contains:

• sfm/model.py
  • SFM: In a directed acyclic graph, the value of a node is determined by the values of its parents through a function.
• sfm/inference.py
  • Vanilla forward inference (VFI): w = VFI(M, w_exo) Given the SFM and the values of all exogenous nodes (root nodes), compute the values of all nodes.
  • Contrastive forward inference (CFI): w1 = VFI(M, w0, w1_exo) Given the SFM, a reference {node: value} assignment over all nodes, and the new values of exo-nodes, compute the values of all nodes. This could reduce the number of function evaluations, since we know from the graph structure that some nodes' values will remain the same.
• sfm/partial.py
  • Partial VFI: w_targets = partial_VFI(M, w_exo, targets) Same input as VFI, except we're only interested in the values of some target nodes. Using graph information, we further reduce the number of function evaluations.
  • Partial CFI: w1_targets = partial_CFI(M, w0, w1_exo, targets) Same input as CFI, except we're only interested in the values of some target nodes. This is the setting with the fewest function evaluations.
• generate.py
  • Generate random directed acyclic graphs by assigning random topological orders over undirected Erdos-Renyi graphs.
  • Generate random SFMs with random linear functions/congruences:
  • f(x1, x2, ...) = a1 * x1 + a2 * x2 + ...
  • f(x1, x2, ...) = a1 * x1 + a2 * x2 + ... (mod m) When all variables/coefficients have integer values, this helps create non-injective functions for testing CFI.