First, I'm from computer science. I don't know how to say this problem in a mathematical way. So please bear with me.
The question
Let say I have a factor graph illustrated in the figure.
The variables are $w,t,a$. The domain for $w,t,a$ is $\{0,1\},\{0,1,2\}$ and $\{0,1,2\}$ respectively. For the factors $h,g$, I know how to assign functions to them. For $h$, it can take a probability density function: $Pr(0) = 0.7, Pr(1)=0.3$. For $g$, it can be $Pr(0) = 0.2,Pr(1) = 0.3, Pr(2) = 0.5$. (Correct me if these assignments are wrong)
Then, I don't know how to assign a function to the factor $f$. Is it the conditional probability distribution function $p(t|w,a)$ ? If this is the case, do I need to define these three functions $p(t=0|w,a),p(t=1|w,a),p(t=2|w,a)$ given all possible values of $w$ and $a$ ?
The setting
The real world setting is the following: A user $w$ needs to label a data item $t$ to its correct value. The values which a data item $t$ can take are $0,1,2$. The value of the user can be $0,1$ - 0 means unreliable and 1 means reliable. $a$ is the label that the user have assigned to the data item. Therefore, I think $a$ is an observed variable and it can only take one value. From the factor graph and these values, I want to find the reliability of the user and the probability of correctness for each value of the data item.
Please enlighten me. Thanks.