I am having trouble computing the remaining probabilities with the use of an interaction model called the noisy-MAX. The noisy-MAX model is an interaction model which helps a network engineer assessing probabilities for the CPT of a Bayesian Network. The noisy-MAX model is defined as follows:
The parameter probabilities for the noisy-MAX model describe for each cause variable separately, the influence of its different manifestation levels on the possible values of the effect variable, that is, the model takes the parameter probabilities $Pr(e^i|c_{1}^{0},...,c_{j-1}^{0}, c_{j}^{k},c_{j+1}^{0},...,c_{n}^{0})$ for all values $c_{j}^{k}, k >0,$ of the cause variable $C_j$ and all values $e^i, i \geq 0,$ of the effect variable $E$. The remaining probabilities for the conditional probability table for $E$ are defined through
$Pr(e^i|\mathbf{c})$ =
\begin{cases}
Pr(E\leq e^i|\mathbf{c})- Pr(E\leq e^{i-1}|\mathbf{c}) & \text{for } i > 0 \\
Pr(E\leq e^0|\mathbf{c}) & \text{for } i = 0
\end{cases}
with
\begin{equation*}
Pr(E\leq e^i|\mathbf{c}) = \prod_{j \in J} \sum_{l=0,...,i} Pr(e^i|c_{1}^{0},...,c_{j-1}^{0}, c_{j}^{k},c_{j+1}^{0},...,c_{n}^{0})
\end{equation*}
where $J$ is the set of indices of the cause $C_j$ that are marked as having a value $c_{j}^{k}$ with $k>0$ in the joint value combination $\mathbf{c}$.
Now, if for example, we assume that the cause variable $C_1$ is ternary and cause variable $C_2$ and effect variable $E$ are binary. We have:
$C_1=\{c_{1}^{0}, c_{1}^{1}, c_{1}^{2}\}$, $C_2=\{c_{2}^{0}, c_{2}^{1}\}$, $E=\{e^{0}, e^{1}\}$
The basic causal mechanism looks like: $(C_1)-->(E)<---(C_2)$
How can I compute, for example: $Pr(e|c_{1}^{1}, c_2^1)$? Or $Pr(e|c_{1}^{2}, c_2^1)$
Values for the parameter probabilities $Pr(e|c_{1}^{1}, c_2^0)$, $Pr(e|c_{2}^{1}, c_2^0)$ and $Pr(e|c_{1}^{0}, c_2^1)$ have been obtained (these probabilities need to be assessed by a domain experts in order to be able to use the interaction model).
I hope someone can help me!
Kind regards.
