Finding the maximum likelihood parameter estimates for a given bayesian network I am studying the book by Adnan Darwiche (Modelling and Reasoning with Bayesian Networks), specifically Chapter 17.

I am unsure about how to proceed with this exercise. Since A leads to B and B leads to C, I know how to compute the maximum likelihood parameters for A leading to B.
For B leading to C, we can write the probability $P(C|B,A) = \dfrac{P(A,B,C)}{P(A) P(B|A)} $. But then, $P(B|A) = \dfrac{P(A,B)}{P(A)}$. This means that $P(C|B,A) = \dfrac{P(A,B,C)}{P(A,B)} $.
Is this correct, how does this work out then?
 A: First of all, because you have the chain $A\to B\to C$, it follows that $C$ is independent of $A$ given $B$, so $P(C|B, A) = P(C|B)$.
From the table, I presume that all your three random variables are boolean. That means you have five parameters in total:
$$p(A, B, C) = p(A) p(B|A) p(C|B),$$
where for the first factor $p(A)$ you need just one parameter (because $p(A=T) = 1-p(A=F)$, and similarly you need 2 parameters each for the second and third factor. Lets give them names:
$$\begin{align}
a   & := & p(A=T)\\
b_T & := & p(B=T | A=T)\\
b_F & := & p(B=T | A=F)\\
c_T & := & p(C=T | B=T)\\
c_F & := & p(C=T | B=F)\\
\mathbf{\theta} & := & (a, b_T, b_F, c_T, c_F)
\end{align}$$
You want to find the MLE of those five parameters $\theta$, i.e. those that give the largest probability to your table. I.e. you want to maximize (provided your data is independent given the parameters $\theta$):
$$o(\theta) = p(T, F, F)^2 p(F, F, T) p(T, F, F),$$
where each of those probabilities is a function of $\theta$, e.g.
$$p(T, F, F) = a (1-b_T) (1-c_F).$$
Thus, $o(\theta)$ is a polynomial of order 12 in $\theta$. Finding the local maxima of $o(\theta)$ is the standard procedure of checking both the set of spots where the gradient vanishes ($grad_\theta o(\theta) = 0$), as well as the boundary. I leave this up to you.
