Independence of Cause and Mechanism - Causality 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?

I would say this is a matter of terminology. What people in causality mean when they talk about mechanisms (also known as Independent Causal Mechanisms, causal Markov kernels, causal conditionals, etc.) are the "true" causal conditional distributions. In your case, you have only one mechanism $P(B \mid A)$.
As you correctly pointed out, the independence of cause and mechanism is just a way of saying that if you intervene on a variable, you don't change the conditional of downstream variables.
I don't know exactly what do you mean by showing this is true, this is a bit of an "axiom" of causality. But if I was forced to do so, then I would start with the causal Markov condition, which states that a causal graph can be decomposed into the "true" conditional distributions.
