Proof of a causual (line) Bayesian graph model Given a simple Bayesian graph model, and $A$ is observed.
A <---- B <---- C

The joint model is
$$
p(A,B,C) = p(A\mid B,C)p(B\mid C)p(C),
$$
which is true. But I don't understand why people use
$$
p(A\mid B,C) = p(A\mid B)
$$
such that
$$
p(A,B,C) = p(A\mid B)p(B\mid C)p(C)
$$
For example, https://frnsys.com/ai_notes/foundations/probabilistic_graphical_models.html  or https://ermongroup.github.io/cs228-notes/representation/directed/ says it uses belief net's conditional independence assumptions.
But why? Are they really equal? How to prove this?
 A: The equation $p(A,B,C) = p(A\mid B,C)p(B\mid C)p(C)$ holds always. You don't need a Bayesian network or any other probabilistic structure. 
And $P(A|B,C)\neq P(A|B)$ in general. They're equal here just because of the Bayesian network's structure. So, you can't prove it. It's by definition. In a Bayesian network, a node is conditionally independent from any other node given its immediate parents. 
A: It's true any 3-variable probability distribution $p(A,B,C)$ can be factored as $p(A|B,C) p(B|C)p(C)$.  If it is the case that $p(A| B, c) = p(A| B, c')$ for all $c,c' \in C$, then we have $p(A|B,C) = p(A|B)$ and $A$ is conditionally independent of $C$ given $B$.  This conditional independence is represented in the graph structure by the lack of an edge from $C \rightarrow A$.
How is the case that $A$ is conditionally indpendent of $C$ (given $B$)?  that is just a particular feature of the particular model at hand.  Sometimes it is just assumed in order to make the model tractable.  Sometimes it is an empirical observation, sometimes it's based on a particular theory etc.
When you're given the graphical model $C \rightarrow B \rightarrow A$ that is a way of expressing the idea that "in this model $A$ is conditionally independent of $C$ given $B$ -- this is an assumption/restriction built into that model.
