PEARL, JUDEA — "CAUSALITY: MODELS, REASONING, AND INFERENCE" (2nd ed., 2009)
Plus supporting paraphrase from Wikipedia on Causal graph and Bayesian network.
NOTE: Verbatim text from Pearl 2009 is not reproduced here. [PARAPHRASE] marks restatements of well-established results.

(a) DAGs ARE FIXED RELATIVE TO A CAUSAL MODEL

A Pearlian causal model M is a triple <U, V, F> together with a directed acyclic graph G whose vertices are the variables in V and whose edges encode the structural functions in F. The graph G is not an optional decoration: it is constitutive of the model. To specify M is to specify G.

[PARAPHRASE, Pearl 2009, Ch. 1 and Ch. 7] Every claim made "in M" — every conditional independence, every identifiability result, every do-calculus derivation — is a claim relative to THIS G. The semantics of intervention (do(X=x)) is defined as graph surgery on G: cut the incoming arrows to X in G and propagate. Without a fixed G, do(.) has no referent.

Wikipedia (Causal graph): "Each variable in the model has a corresponding vertex or node and an arrow is drawn from a variable X to a variable Y whenever Y is judged to respond to changes in X when all other variables are being held constant." The arrow set IS the model.

(b) ADDING AN EDGE REPLACES G WITH G' — OLD THEOREMS NO LONGER HOLD

This is not a stylistic point; it is a logical one. If G has edge set E and we add a new edge e to obtain G' with edge set E ∪ {e}, then G ≠ G'. Any theorem of the form "in G, X ⫫ Y | Z" was a theorem about G's d-separation structure. The new graph G' has a different d-separation structure: in particular, the new edge can open paths that were previously blocked, destroying conditional independences that the old theorems relied on.

[PARAPHRASE] Consequences:
 - Identifiability results proved in G can FAIL in G'. A causal effect that was identifiable from observational data in G may become non-identifiable once an edge is added.
 - Markov equivalence classes change. Two DAGs are Markov-equivalent iff they share the same skeleton and v-structures; adding an edge generally moves G into a different equivalence class.
 - Adjustment sets change. A back-door admissible set Z for the effect of X on Y in G is not guaranteed to remain admissible in G'.
 - The structural equations F must also be updated; F was indexed by parents-in-G, and a new edge introduces a new parent that the old F_Y did not range over.

The methodological rule (Pearl, repeatedly): you cannot prove a theorem in one graph and then "extend" the graph and reuse the theorem. You must reprove it in the extended graph, or show that the relevant property is preserved.

Application note for debate: any argument of the shape "the BIT framework's DAG has property P (proved in G); we now add an edge for agent-driven selection f; therefore the framework still has property P" is invalid by Pearl's own ground rules. The addition of f either (i) changes G, in which case P must be reproved, or (ii) does NOT change G, in which case f is doing no causal work in the model.

(c) THE d-SEPARATION FRAMEWORK

d-separation is the graphical criterion for conditional independence in a DAG. A path between X and Y is BLOCKED by a set Z iff:
 1. it contains a chain A -> M -> B or a fork A <- M -> B with M ∈ Z, OR
 2. it contains a collider A -> M <- B such that M ∉ Z and no descendant of M is in Z.
X and Y are d-separated by Z iff every path between them is blocked by Z. Pearl's theorem (and a foundational result of the Bayesian network literature): if X and Y are d-separated by Z in G, then X ⫫ Y | Z in every distribution that factorizes according to G.

The contrapositive matters for debate: d-separation is RELATIVE TO G. Change G, and you change which (X,Y,Z) triples are d-separated. There is no graph-free notion of d-separation in Pearl's framework.

Wikipedia (Bayesian network) confirms: "the network's graphical structure is typically established beforehand... Once fixed, this DAG architecture determines all conditional independence relationships in the system." The conditional-independence content of a Bayesian network is exactly what its (fixed) DAG says it is.
