Adjustment Identification Distance: A 𝚐𝚊𝚍𝚓𝚒𝚍 for Causal Structure Learning

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.

If you publish research using 𝚐𝚊𝚍𝚓𝚒𝚍, please cite our UAI paper

@inproceedings{henckel2024adjustment,
    title = {{Adjustment Identification Distance: A gadjid for Causal Structure Learning}},
    author = {Leonard Henckel and Theo Würtzen and Sebastian Weichwald},
    booktitle = {{Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence (UAI)}},
    year = {2024},
    doi = {10.48550/arXiv.2402.08616},
}

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Get Started Real Quick 🚀

𝚐𝚊𝚍𝚓𝚒𝚍 is implemented in Rust and can conveniently be installed from PyPI (pip install gadjid) to be used from Python and from CRAN (install.packages("gadjid")) to be used from R.

Installation – Python

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

Install Alternatives

Pip tries to find a matching wheel and install that. Since we offer precompiled wheels for most common operating systems, python versions, and CPU architectures, the installation will usually be quick. If there is no matching wheel (or when explicitly installing from source via pip install gadjid --no-binary gadjid), pip will download the source distribution and compile a wheel for the current platform, which requires the rust toolchain to be installed.

The current development version can be compiled and installed via
pip install "git+https://github.com/CausalDisco/gadjid.git"
or by cloning this repository and calling either
maturin develop --manifest-path ./gadjid_python/Cargo.toml (unoptimized dev compile) or
maturin develop --manifest-path ./gadjid_python/Cargo.toml --release (optimized release compile).

Introductory Example – Python

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))

Installation – R

install.packages("gadjid")

A source package install requires the rust toolchain to be installed.

Introductory Example – R

library(gadjid)

?ancestor_aid

g_true <- rbind(c(0, 1, 1, 1),
                c(0, 0, 1, 1),
                c(0, 0, 0, 1),
                c(0, 0, 0, 0))
g_guess <- rbind(c(0, 1, 0, 0),
                 c(0, 0, 1, 0),
                 c(0, 0, 0, 1),
                 c(0, 0, 0, 0))

ancestor_aid(g_true, g_guess, edge_direction = "from row to column")
oset_aid(g_true, g_guess, edge_direction = "from row to column")
parent_aid(g_true, g_guess, edge_direction = "from row to column")
shd(g_true, g_guess)
sid(g_true, g_guess, edge_direction = "from row to column")

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 UAI paper).

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 |