Identification

Once the graph is restricted to the edges that earn an arrow, the question becomes whether a causal effect can be computed from observational data at all.

The criterion

A set Z satisfies the backdoor criterion relative to (X, Y) when

1. no node in Z is a descendant of X, and
2. Z blocks every path between X and Y that starts with an arrow into X

(Pearl, Causality, 2nd ed., Def. 3.3.1).

Both halves matter. The descendant condition rules out conditioning on a mediator or a collider downstream of the treatment. And blocking only the backdoor paths — never the causal path itself — is what a plain d-separation test does not give you for free.

How the second half is tested

To use a generic d-separation routine, the edges leaving the treatment are removed first. That deletes the causal path, and anything X causes, while leaving every backdoor path exactly as it was.

This is not the same mutilation as mutilated_graph, which removes the edges into the treatment to build the interventional graph G_X̄. The two are easy to confuse and mean opposite things.

One call

from causalphylo import find_adjustment_set

verdict = find_adjustment_set(G, treatment="X", outcome="D")

find_adjustment_set takes the full graph and does the restriction itself — causal_subgraph, then acyclic_core — so no caller can skip it. It returns a frozen Identification:

Field	Meaning
identifiable	Whether the effect is identifiable
adjustment_set	The minimal valid set, as a tuple
reason	An IdentificationReason, always set
search_truncated	Whether the subset search hit its bound
n_near_miss_confounders	Confounders the graph hints at but does not license

The reasons

import networkx as nx
from causalphylo import find_adjustment_set

REG = {"edge_type": "activates",       "evidence_class": "regulatory"}
GEN = {"edge_type": "associated_with", "evidence_class": "genetic_association"}
LIT = {"edge_type": "associated_with", "evidence_class": "literature"}

identifiable_by_adjustment

A confounder is present and adjustable.

G = nx.DiGraph()
G.add_edge("U", "X", **REG); G.add_edge("U", "D", **GEN); G.add_edge("X", "D", **GEN)
find_adjustment_set(G, "X", "D")
# identifiable=True  adjustment_set=('U',)  reason='identifiable_by_adjustment'

identifiable_trivially

No confounder appears in the graph, so the empty set satisfies the criterion.

G = nx.DiGraph()
G.add_edge("X", "D", **GEN); G.add_edge("W", "X", **REG); G.add_edge("W", "D", **LIT)
find_adjustment_set(G, "X", "D")
# identifiable=True  adjustment_set=()  reason='identifiable_trivially'
# n_near_miss_confounders=1

A trivial verdict is not a clean bill of health

A knowledge graph is open-world. A missing edge is missing knowledge, not evidence of no confounding.

So a trivial verdict carries n_near_miss_confounders: how many nodes reach the treatment and carry some edge to the outcome that fell short of admissibility. Above, W regulates X and is associated with D — but only by literature, so it is exactly the kind of confounder the graph hints at and cannot license.

A target with many near misses is fragile. This is the same move as an E-value in observational epidemiology: not "there is no confounding", but "here is how much it would take to break this".

no_causal_path

No admissible directed path from treatment to outcome. An association on its own produces this, because it never became an arrow.

G = nx.DiGraph(); G.add_edge("X", "D", **LIT)
find_adjustment_set(G, "X", "D")
# identifiable=False  reason='no_causal_path'

cyclic_component

Treatment or outcome sits in a feedback loop, where d-separation is undefined.

G = nx.DiGraph()
G.add_edge("X", "D", **GEN); G.add_edge("X", "Y", **REG); G.add_edge("Y", "X", **REG)
find_adjustment_set(G, "X", "D")
# identifiable=False  reason='cyclic_component'

no_valid_adjustment_set

Confounders exist, but no admissible subset blocks every backdoor path.

treatment_absent / outcome_absent

The node is not in the admissible subgraph — it may exist in your graph and have had no admissible edge.

G = nx.DiGraph(); G.add_edge("X", "D", **GEN)
find_adjustment_set(G, "ZZZ", "D")   # reason='treatment_absent'
find_adjustment_set(G, "X", "ZZZ")   # reason='outcome_absent'

Search bounds, reported not hidden

The subset search is bounded, because the number of candidate subsets grows as C(pool, size):

>>> from causalphylo import MAX_ADJUSTMENT_SET_SIZE, MAX_CANDIDATE_POOL
>>> MAX_ADJUSTMENT_SET_SIZE, MAX_CANDIDATE_POOL
(3, 20)

Reaching a bound sets search_truncated. It is never silently swallowed:

G = nx.DiGraph(); G.add_edge("X", "D", **GEN)
for i in range(25):                      # 25 confounders, pool bound is 20
    G.add_edge(f"U{i}", "X", **REG); G.add_edge(f"U{i}", "D", **GEN)

find_adjustment_set(G, "X", "D")
# identifiable=False  reason='no_valid_adjustment_set'  search_truncated=True

A False with search_truncated=True means not found within the bound — not does not exist. Treat the two differently.

Working at a lower level

from causalphylo import causal_subgraph, acyclic_core, satisfies_backdoor

dag = acyclic_core(causal_subgraph(G)).dag

satisfies_backdoor(dag, "X", "D", frozenset({"U"}))   # True
satisfies_backdoor(dag, "X", "D", frozenset())        # False

satisfies_backdoor expects a DAG containing both nodes, which find_adjustment_set guarantees before calling it. Reaching for it directly means taking on that guarantee yourself.