If I have a sequence of three random variables let's say $A$, $B$, $C$ such that $B$ only depends on $A$ and $C$ only depends on $B$ the goal is to find an expression for the joint probability of all three variables $P(A, B, C)$.
This is clearly a Markov process and my intuition tells me that you should get $P(A, B, C) = P(A)P(B|A)P(C|B)$ but I can't find my way to this result. If we use the following expression for the conditional probability:
$$ P(X|Y)=\frac{P(X, Y)}{P(Y)} $$
Then
$$ \begin{align} P(A, B, C) = P(C, B, A) = P(C|B, A)P(B,A) &= P(C|B, A)P(B|A)P(A) \end{align} $$
The term $P(C|B, A)$ should be equal to P(C|B) because $C$ does not care about the results on $A$ but I can't prove. Any tips ? Am I completely wrong here ?
In a Marcov chain with a DAG of $A\to B\to C$, the events $A$ and $C$ are conditionally independent when given $B$. This is the structure of dependencies you have described. Therefore $\mathsf P(A,C\mid B)=\mathsf P(A\mid B)~\mathsf P(C\mid B)$, so...
$$\begin{align}\mathsf P(A,B,C)&=\mathsf P(B)~\mathsf P(A,C\mid B)\\[1ex]&=\mathsf P(B)~\mathsf P(A\mid B)~\mathsf P(C\mid B)\\[1ex]&=\mathsf P(A)~\mathsf P(B\mid A)~\mathsf P(C\mid B)\end{align}$$