I'm currently going through Prof. Daphne Koller's probabilistic graphical models course on Coursera and had a question regarding an exercise problem.

The problem is as follows:

How many independent parameters are required to uniquely define the conditional probability distribution of $C$ in the graphical model below, if $A$, $B$, and $D$ are binary, and $C$ and $E$ have three values each?

enter image description here

My approach is that since $C$ is dependent on $A$ and $B$ (i.e., the CPD of $C$ is $P(C\ \vert\ A, B)$) and $A$, $B$, and $C$ each have 2, 2, and 3, values, respectively, we can write $2 \times 2 \times 3$. But we have to subtract $1$ since knowing all but the last values of $P(C\ \vert\ A, B)$ automatically determines the last. Therefore, we get $(2 \times 2 \times 3) - 1 = 11$ independent parameters.

Where am I going wrong? Any tips are appreciated, thanks.


1 Answer 1


Given the values of $A$ and $B$, there are $2\times2=4$ different conditional probability functions of $C$. For example, if $A=1,B=1$: $$P(C=c_1|A=1,B=1)+P(C=c_2|A=1,B=1)+P(C=c_3|A=1,B=1)=1$$

So, if you happen to know any two of these probabilities, you can find the third because they sum up to $1$. For each of these four conditional probabilities, you need to know $2$ parameters, ending up with a total of $8$ parameters.


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