Is there a way to get a factorization of this function with respect to a particular variable?
What I mean by this is basically not to get a unique global factorization where you have products of terms, but if you can have a function to generate a factorization based on a particular variable (like factorizing with respect to p1 for example)?
The reason why I suggest this is because I suspect that factorizing with respect to a specific variable should always give the same form regardless of the variable.
As an example with three probabilities you have p1 + p2 + p3 - p1p2 - p1p3 - p2p3 + p1p2p3 gives with respect to the three variables:
p1(1 - p2 - p3 + p2p3) + p2 + p3 - p2p3
p2(1 - p1 - p3 + p1p3) + p2 + p3 - p1p3
p3(1 - p1 - p2 + p1p2) + p1 + p2 - p1p2
This helps immensely in the Lagrange multiplier process when it comes to doing derivations for n-variables. If you can show that the form is the same for every kind of factorization, then when you use the Lagrange technique, you should get a bunch of these terms that are all similar.