The example given in the text is of the relationship between a fire alarm, smoke, and fire. 

Fire causes smoke, which causes the fire alarm to go off. The three links are necessary in the graph because fire does not directly cause the fire alarm to go off. Therefore if we know that there is smoke, the presence of fire does not lift nor lower the probability of hearing the fire alarm go off. 

Example of the Screening Off Effect

The mediator is a variable that functions as the mechanism by way of transferring an effect of one thing to another.  

Consider the chain junction, where $A -> B -> C$.  $B$ is the mediator that transfers the effect of $A$ to $C$.   In other words, the mediator, $B$, is the variable that explains the causal effect of $A$ on $C$.  $A$ can be an indirect cause of $C$, meaning that $A$ affects $B$, which in turn contributes to the change in $C$.  

Mediator

When the junction $$A \rightarrow B \rightarrow C$$ appears in a causal diagram, we note that the presence of $$B$$ "screens-off" the value of $$A$$. This means that $$C$$ and $$A$$ are conditionally independent given the value of $$B$$.

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Pearl outlines three basic types of junctions in causal diagrams which appear commonly and help explain the data. The three junctions take the forms: 1. $$A \rightarrow B \rightarrow C$$ (Chain) 2. $$A \leftarrow B \rightarrow C$$ (Fork) 3. $$A \rightarrow B \leftarrow C$$ (Collider)

Three Basic Junctions in Causal Diagrams

Pearl, J., & Mackenzie, D. (2018). $$\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$$ Basic Books.

The Book of Why

When the junction $A \leftarrow B \rightarrow C$ appears in a Causal Diagram, we say that B is a confounder for both A and C. It might appear that there is a causal relationship between A and C, because there is an observable relationship between A and C when B is varied. However, if we condition on B ("hold B constant") we observe that A and C become conditionally independent. 

$A \leftarrow B \rightarrow C$: The Fork and Confouders

When a single node is caused by two other nodes we encounter something called "collider-bias" where one condition explains-away another. 

Since A and C both effect the outcome of B, by conditioning on B, A and C become **dependent**.

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