I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$. I am able to write each event as a sum of distinct events that form a partition of the space.
My goal is to find all the relationships between the probability of the events. So I am trying to find all relationships like:
- $\sum \lambda _i P_i = T$
- $P_i \le P_j$ and more generically $\sum \lambda_i P_i * \sum \lambda_j P_j \le T$
- $P_i P_j = P_l$ and more generically $\sum \lambda_i P_i * \sum \lambda_j P_j = T$
- and other relationships (maybe using division, etc)
I have been able to solve the first type of relationships. Here's my process. I write the matrix of my events. Each column is an event, each row represents the realisation of the distinct event of my partition. So I get a boolean matrix that would look like this if I had 3 events, and a partition of 4 elements. \begin{bmatrix} 1 & 1 & 1\\ 0 & 1 & 1\\ 1 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix} Then calculating the kernel of that matrix gives me the relationship between the $(A_i)_i$, which are random variable that have a value of 1 if $A_i$ is true, 0 otherwise. The kernel gives me the linear combinations that sum to 0 of the $A_i$. By taking the expected value of that linear combination, I get the linear combinations between the probabilities. To solve it with $T \neq 0$, it is simply solving a linear equation
For the second category of relationships (inequalities, that is inclusion of events), does anyone know a good way to solve this (If you could also point me to an implementation, maybe in R or python, then that'd be great!). It seems to be in the category of "Linear matrix Inequality", on which I have no experience.
For the other category ($And$ relationships), any thoughts on how to solve this?
Generically, I feel I might be missing something. Is there a field of mathematics that addresses this question. Am I taking the wrong approach? Any suggestion very welcome. Thanks a lot!