2
$\begingroup$

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.

$\endgroup$

1 Answer 1

1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.