gadjid

Adjustment Identification Distance: A 𝚐𝚊𝚍𝚓𝚒𝚍 for Causal Structure Learning

This is an early release of 𝚐𝚊𝚍𝚓𝚒𝚍 🐥 and feedback is very welcome! Just open an issue on github.

If you publish research using 𝚐𝚊𝚍𝚓𝚒𝚍, please cite our article

@article{henckel2024adjustment,
    title = {{Adjustment Identification Distance: A gadjid for Causal Structure Learning}},
    author = {Leonard Henckel and Theo Würtzen and Sebastian Weichwald},
    journal = {{arXiv preprint arXiv:2402.08616}},
    year = {2024},
    doi = {10.48550/arXiv.2402.08616},
}

Get Started Real Quick 🚀 – Introductory Example

Just pip install gadjid to install the latest release of 𝚐𝚊𝚍𝚓𝚒𝚍
and run python -c "import gadjid; help(gadjid)" to get started (or see install alternatives).

import gadjid
from gadjid import example, ancestor_aid, oset_aid, parent_aid, shd
import numpy as np

help(gadjid)

example.run_parent_aid()

Gtrue = np.array([
    [0, 1, 1, 1, 1],
    [0, 0, 1, 1, 1],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0]
], dtype=np.int8)
Gguess = np.array([
    [0, 0, 1, 1, 1],
    [1, 0, 1, 1, 1],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0]
], dtype=np.int8)

print(ancestor_aid(Gtrue, Gguess, edge_direction="from row to column"))
print(shd(Gtrue, Gguess))

---

𝚐𝚊𝚍𝚓𝚒𝚍 is implemented in Rust and can conveniently be called from Python via our Python wrapper (implemented using maturin and PyO3).

Evaluating graphs learned by causal discovery algorithms is difficult: The number of edges that differ between two graphs does not reflect how the graphs differ with respect to the identifying formulas they suggest for causal effects. We introduce a framework for developing causal distances between graphs which includes the structural intervention distance for directed acyclic graphs as a special case. We use this framework to develop improved adjustment-based distances as well as extensions to completed partially directed acyclic graphs and causal orders. We develop new reachability algorithms to compute the distances efficiently and to prove their low polynomial time complexity. In our package gadjid, we provide implementations of our distances; they are orders of magnitude faster with proven lower time complexity than the structural intervention distance and thereby provide a success metric for causal discovery that scales to graph sizes that were previously prohibitive.

Parallelism – setting the number of threads

𝚐𝚊𝚍𝚓𝚒𝚍 uses rayon for parallelism using, per default, as many threads as there are physical CPU cores. The number of threads to use can be set via the environment variable RAYON_NUM_THREADS. We recommend to do so and to set the number of threads manually, not least to be explicit and to avoid the small runtime overhead for determining the number of physical CPU cores.

Implemented Distances

• ancestor_aid(Gtrue, Gguess, edge_direction)
• oset_aid(Gtrue, Gguess, edge_direction)
• parent_aid(Gtrue, Gguess, edge_direction)
• for convenience, the following distances are implemented, too
  • shd(Gtrue, Gguess)
  • sid(Gtrue, Gguess, edge_direction) – only for DAGs!

where Gtrue and Gguess are adjacency matrices of a DAG or CPDAG and edge_direction determines whether a 1 at r-th row and c-th column of an adjacency matrix codes the edge r → c (edge_direction="from row to column") or c → r (edge_direction="from column to row"). The functions are not symmetric in their inputs: To calculate a distance, identifying formulas for causal effects are inferred in the graph Gguess and verified against the graph Gtrue. Distances return a tuple (normalised_distance, mistake_count) of the fraction of causal effects inferred in Gguess that are wrong relative to Gtrue, normalised_distance, and the number of wrongly inferred causal effects, mistake_count. There are $p(p-1)$ pairwise causal effects to infer in graphs with $p$ nodes and we define normalisation as normalised_distance = mistake_count / p(p-1).

You may also calculate the SID between DAGs via parent_aid(DAGtrue, DAGguess, edge_direction), but we recommend ancestor_aid and oset_aid and for CPDAG inputs the parent_aid does not coincide with the SID (see also our accompanying article).

If edge_direction="from row to column", then a 1 in row r and column c codes a directed edge r → c; if edge_direction="from column to row", then a 1 in row r and column c codes a directed edge c → r; for either setting of edge_direction, a 2 in row r and column c codes an undirected edge r – c (an additional 2 in row c and column r is ignored; one of the two entries is sufficient to code an undirected edge).

An adjacency matrix for a DAG may only contain 0s and 1s. An adjacency matrix for a CPDAG may only contain 0s, 1s and 2s. DAG and CPDAG inputs are validated for acyclicity. However, for CPDAG inputs, the user needs to ensure the adjacency matrix indeed codes a valid CPDAG (instead of just a PDAG).

Empirical Runtime Analysis

Experiments run on a laptop with 8 GB RAM and 4-core i5-8365U processor. Here, for a graph with $p$ nodes, sparse graphs have $10p$ edges in expectation, dense graphs have $0.3p(p-1)/2$ edges in expectation, and x-sparse graphs have $0.75p$ edges in expectation.

Maximum graph size feasible within 1 minute

Method	sparse	dense
Parent-AID	13601	962
Ancestor-AID	8211	932
Oset-AID	1105	508
SID in R	256	239

Results obtained with 𝚐𝚊𝚍𝚓𝚒𝚍 v0.1.0 using the Python interface and the SID R package v1.1 from CRAN.

Average runtime

Method	x-sparse ($p=1000$)	sparse ($p=256$)	dense ($p=239$)
Parent-AID	7.3 ms	30.5 ms	173 ms
Ancestor-AID	3.4 ms	40.9 ms	207 ms
Oset-AID	5.0 ms	567 ms	1.68 s
SID in R	~1–2 h	~60 s	~60 s

Results obtained with 𝚐𝚊𝚍𝚓𝚒𝚍 v0.1.0 using the Python interface and the SID R package v1.1 from CRAN.

LICENSE

𝚐𝚊𝚍𝚓𝚒𝚍 is available in source code form at https://github.com/CausalDisco/gadjid.

This Source Code Form is subject to the terms of the Mozilla Public License, v. 2.0. If a copy of the MPL was not distributed with this file, You can obtain one at https://mozilla.org/MPL/2.0/.

See also the MPL-2.0 FAQ.