I have an output binary scalar, $y∈B=\{0,1\}$, and an input binary vector $x=[x_1, x_2,…x_M]$ where $x_i∈B=\{0,1\}$. I know that the output $P(y)=1$ depends entirely on the input x. Thus, I want to simplify the expression:
$ Eq.1: P(y|x_1∩x_2∩…x_M) $
I know 2 quantities:
$ Eq. 2a: P(y|x_i) $
$ Eq. 2b: P(y|x_i∩x_j) $
For all i & j
Furthermore, I can make 2 assumptions:
- The input, x, is a ‘white’ / Poisson process. Ie, $x_i$ and $x_j $ are uncorrelated for all $i\neq j$.
- The output y is dependent only on 1st and 2nd order statistics of the inputs shown in eq. 2a,b. Ie, single pulses and pairs of pulses affect y, while triplets of input pulses have no effect on y. I am not sure what the exact mathematical term for this is...
Thus, my question is can I use the above knowledge and assumptions to simplify equation 1 and hopefully solve for it, preferably in terms of the 1st and 2nd order quantities