By causal direction (X causes Y) I mean that in the induced graph of the structural causal model, X appears before Y. The graph induced by a structural causal model has an edge from X to Y if there is a structural equation Y = F(U_i, X, β¦ ).
It refers to Example 9.3 and the associated lemmas.
This example is a general rephrasing of Example 1 of finite factored sets in terms of random variables and structural causal models.
Example 9.3
We observe two nonconstant, (real valued) random variables V1 and V2. We assume that they are part of an unobserved true underlying causal model (π, π , πΉ, β), where π = (Vi)iβI and {1, 2} β πΌ. We only observe the distribution of π . Let ZβV1+V2. Suppose that V1β««PZ. Furthermore, we assume that β is in βgeneral positionβ. This means that this independence is not due to the specific choice of β. Formally, V1β««PZ for all causal models (π, π , πΉ, π) s.t. π βΌ β.
It says that any structural causal model that models the independence of V1 and Z structurally (it is not a coincidence of precise numerical values of β), then in this structural causal model has V1βV2, i.e. in the graph induced by the causal model through itβs equation, V1 always is an ancestor of V2 .
What is meant by βcausal directionβ ? Where in the paper is this referring to?
By causal direction (X causes Y) I mean that in the induced graph of the structural causal model, X appears before Y. The graph induced by a structural causal model has an edge from X to Y if there is a structural equation Y = F(U_i, X, β¦ ).
It refers to Example 9.3 and the associated lemmas.
This example is a general rephrasing of Example 1 of finite factored sets in terms of random variables and structural causal models.
It says that any structural causal model that models the independence of V1 and Z structurally (it is not a coincidence of precise numerical values of β), then in this structural causal model has V1βV2, i.e. in the graph induced by the causal model through itβs equation, V1 always is an ancestor of V2 .