This is my problem
My problem is modeled by a basic Bayesian Network with only two layers. So I have parent and child nodes but the children has no children. Essentially a bipartite graph. The children depend on one or more of the parents and these edges/relations have a probability. The model is also extended by completing the bipartite graph with low probability edges to account for noisy data. So in the finished model each child $(C_x)$ has a dependency on each parent $(P_x)$.
What I want to be able to do is to infer the most probable solution by maximizing,
$$ max_{P_1,P_2,\dots,P_n}(P(P_1,P_2,\dots,P_n|C_1,C_2,\dots,C_m)) $$ where $ C_i \in \{0,1\} $, $ P_j \in \{0,1\} $. For a given observation on the state of the children.
So for a observation $O=\{C_1=0, C_2=1, C_3=1\}$
I want to calculate $$P(P_1=1,P_2=0,P_3=0|C_1=0, C_2=1, C_3=1)$$ $$P(P_1=0,P_2=1,P_3=0|C_1=0, C_2=1, C_3=1)$$ $$P(P_1=1,P_2=1,P_3=0|C_1=0, C_2=1, C_3=1)$$ and so on.
Here is how I try to solve it
I want to be thorough so I will explain how I try to solve this. I'm not sure that I'm doing it right. Please point out to med if I'm doing this wrong.
What I do first is I complete the conditional distribution tables for the child nodes. From the model I have the CPT as for a child node $C_x$ something like this, the probabilites are chosen to make the example easy.
\begin{array}{ l l l|l l } P_1 & P_2 & P_3 & P & \lnot P \\ \hline 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0.1 & 0.9 \\ 0 & 1 & 0 & 0.4 & 0.6 \\ 1 & 0 & 0 & 0.2 & 0.8 \\ \end{array}
And I complete this table by calculating
\begin{array}{ l l l|l l } P_1 & P_2 & P_3 & P & \lnot P \\ \hline 0 & 1 & 1 & (1- \lnot P) & 0.9 \times 0.6 \\ 1 & 0 & 1 & (1- \lnot P) & 0.9 \times 0.8 \\ 1 & 1 & 0 & (1- \lnot P) & 0.6 \times 0.8 \\ 1 & 1 & 1 & (1- \lnot P) & 0.9 \times 0.6 \times 0.8 \\ \end{array}
And then I calculate the joint conditional probability by doing something like
$$P(P_1=1,P_2=0,P_3=0|C_1=0, C_2=1, C_3=1) = $$ $$P(P_1=1,P_2=0,P_3=0|C_1=0) \times$$ $$P(P_1=1,P_2=0,P_3=0|C_2=1) \times$$ $$P(P_1=1,P_2=0,P_3=0|C_3=1)$$
Basically I'm not convinced that the last step is correct. Any help or comments is greately appreciated!
Yes, this is school related but it is not regular homework. It is a part of a thesis project I'm doing at a company.
Graph update on request
Here is a graph example of my model. The regular edges are direct dependencies and the dotted edges are noisy or guess edges added to allow for noisy data. Guess edges has a low probability of $p = 0.0001$ and the regular edges all have the probability $(1-p)$. In my problem I do not care for the probability of individual events like $P(L_1)$ or $P(C_2)$, so they are assumed to be $1$. I make an observation on the state of the variables $C_1...C_n$ and given the graph model above I want to infer to most plausible cause. The possible causes are $P_1...P_n$ or a combination of them that is most likely. Like I stated earlier on the example observation above.
Here is an example of the basic truth tables for this graph. \begin{array}{ l|l l } C_1 & T & F \\ \hline P_1 & 0.9999 & 0.0001 \\ P_2 & 0.9999 & 0.0001 \\ P_3 & 0.9999 & 0.0001 \\ \end{array}
\begin{array}{ l|l l } C_2 & T & F \\ \hline P_1 & 0.0001 & 0.9999 \\ P_2 & 0.9999 & 0.0001 \\ P_3 & 0.9999 & 0.0001 \\ \end{array}
\begin{array}{ l|l l } C_3 & T & F \\ \hline P_1 & 0.0001 & 0.9999 \\ P_2 & 0.0001 & 0.9999 \\ P_3 & 0.9999 & 0.0001 \\ \end{array}
Does that make it any clearer?
