In causal modelling, we say that $A \longrightarrow B$ if forcing a value change A will influence the likelihood of $B$ while holding all other variables in the system constant. We call this a direct effect. A consequence of this is "independence of cause and mechanism", i.e. $P(A)$ does not influence the conditional $P(B|A)$.
The way I understand it, is that $A$ needs to be fixed in order to observe the cause of $A$ on $B$, and as $A$ is deterministic $P(B|A)$ is independent of the marginal. Is this correct?
Question: What exactly means "independence of cause and mechanism". How to show that this is true?