Skeleton Discovery

The first stage of constraint-based causal discovery involves skeleton discovery. The F2_CI algorithm will use FCI’s skeleton discovery procedure on _PXm. I therefore analyze the properties of FCI’s skeleton discovery procedure when using _PXm.

Denote the possible d-separating set of Oi at t ∈ S(f(T)) as P DS(Oti). We have

O_jt ∈P DS(Ot_i) if and only if there exists a pathπt betweenOt_i and Ot_j such that, for every subpath hOt

m, Otl, Othi of πt, Olt is a collider on the subpath or hOtm, Olt, Ohti is a triangle.

Next, denote the possible d-separating set of Oi asP DS(Oi), where Oj ∈P DS(Oi) if and

only if Ot

j ∈P DS(Oit) at some t ∈S(f(T)).

Now recall that PC’s skeleton discovery and v-structure orientation procedures discover a superset ofP DS(O_it),∀iusing_PXt under d-separation faithfulness with respect to _PXt. I want to show that PC’s skeleton discovery and v-structure orientation procedures will also discover a superset of P DS(Oi),∀iusing PXm under d-separation faithfulness with respect to_PXt,∀t∈S(f(T)).

Consider the graph Ge constructed by running PC’s skeleton discovery and v-structure orientation procedures over O with a CI oracle. Let Ok ∈ P DS^(Oi) if and only if there

exists a path _eπ in Ge between Oi and Ok such that, for every subpath hOm, Ol, Ohi of eπ,

Ol is a collider on the subpath or hOm, Ol, Ohi is a triangle according to PC’s skeleton

discovery and v-structure orientation procedures. The following lemma shows that we can compute a superset ofP DS(Oi),∀iusing PC’s skeleton discovery and v-structure orientation

procedures:

Lemma 5. Under mixture faithfulness and d-separation faithfulness with respect to_PXt,∀t∈

S(f(T)), PC’s skeleton discovery and v-structure orientation procedures will discover a su- perset of P DS(Oi),∀i with a CI oracle; i.e., P DS(Oi)⊆P DS^(Oi),∀i.

Proof. I need to show that, for any arbitraryOk ∈P DS(Oi), we also haveOk∈P DS^(Oi).

This means that whenever ∃t ∈ S(f(T)) with a path π_kt between O_it and Ot_k such that, for every subpath hOt

m, Olt, Othi of πt, Olt is a collider on the subpath or hOtm, Otl, Othi is a

triangle, then there also exists a path π_ek inGe between O_i and O_k such that O_l is a collider onhOm, Ol, Ohi orhOm, Ol, Ohi is a triangle.

I first show that all adjacencies on πt

k are oneπk. Suppose not. LetO

t

p∗—∗Oqt denote an

arbitrary adjacency present in π_kt such that Op and Oq are not adjacent in eπk. Then, there exists a conditional independenceOp ⊥⊥Oq|(W,S) whenOtp 6⊥⊥Oqt|(Wt,St) by d-separation

faithfulness. But this violates the mixture faithfulness assumption via the contrapositive of Lemma 4. Hence, every adjacency on πt

k is also on eπk.

I next show that ifOt_l is a collider on the subpathhO_mt , O_lt, Ot_hionπ_kt, thenOlis either (1)

a collider on the unshielded subpath hOm, Ol, Ohior (2) hOm, Ol, Ohi is a triangle. Suppose

first that Ol is on the unshielded subpath, so that Om and Oh are non-adjacent in Ge; in other words, Om ⊥⊥Oj|(W,S) for some W ⊆O\ {Oi, Oj}. We already know that Om and

Ol as well as Ol and Oh are adjacent in Ge by the previous paragraph. Now assume that

Ol ∈W for a contradiction. If Olt is a collider on the subpath hOmt , Olt, Ohti on πkt, then we

may invoke Lemma 2.4 in [Colombo et al., 2012] to conclude that there exists an inducing path between Ot

m and Olt that is into Olt and there exists an inducing path betweenOth and

O_lt that is into O_lt. We may now invoke Lemma 2.5 in [Colombo et al., 2012] with the two inducing paths to conclude thatOt

m andOht must be d-connected given{Wt,St}, so we have

by applying the contrapositive of Lemma4; we have arrived at a contradiction. We therefore conclude that Ol 6∈ W, so Ol must be a collider on the unshielded subpath in Ge. Finally, suppose that Ol is on the shielded subpath, so that Om and Oh are adjacent in Ge. Then (2) holds because all three vertices in hOm, Ol, Ohi must be adjacent again by the previous

paragraph.

Recall that P DS(Ot

i) and P DS(Ojt) are sufficiently large sets for determining the ex-

istence of an inducing path between O_it and Ot_j. Thus, we may claim that P DS^(Oi) and

^

P DS(Oj) are also sufficiently large for determining the existence of an inducing path be-

tween O_it and O_jt for any t∈S(f(T)) by invoking Lemma 5.

We may therefore consider the following modified edge interpretations for the skeleton: • The presence of an edge between two vertices Oi and Oj implies that Oi 6⊥⊥ Oj|(W,S)

for all W ⊆P DS^(Oi) and for all W ⊆ P DS^(Oj). Thus, there may exist an inducing

path between Ot

i and Otj at some t∈S(f(T)) in this case by Lemma1.

• The absence of an edge between two vertices Oi and Oj implies that Oi ⊥⊥ Oj|(W,S)

for some W ⊆O\ {Oi, Oj}. Thus, there cannot exist an inducing path betweenOit and

Ot_j at any t∈S(f(T)) in this case by Lemma4.

In other words, I can use FCI’s skeleton discovery procedure equipped with a CI oracle over

PXm to rule out the existence of inducing paths.