V-structure

In the context of a Probabilistic graphical model, a V-structure is a trio of nodes $$A, B, C$$ such that $$A$$ and $$B$$ both influence $$C$$:



V-structures are critical for identifying conditional independence or for d-separated nodes. Similarly, they are used in finding Active trails.

Often, we talk about activating v-structures, which happens when we observe the variable $$C$$. The consequences of observing $$C$$ are important and counterintuitive: The variables $$A$$ and $$B$$, which are originally independent, become dependent when $$C$$ is observed. Formally, we have that

$$P(A,B) = P(A)P(B) \\ P(A,B|C) \neq P(A|C)P(B|C) $$