0
$\begingroup$

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:

  1. The input, x, is a ‘white’ / Poisson process. Ie, $x_i$ and $x_j $ are uncorrelated for all $i\neq j$.
  2. 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

$\endgroup$
4
  • 1
    $\begingroup$ When you say a binary variable is in $[0,1]$, do you mean the set containing $0$ and $1$ rather than the interval from $0$ to $1$? Also, why is this tagged "open-problem?" $\endgroup$ Jan 27, 2014 at 20:16
  • $\begingroup$ Hi Douglas, yes, I mean the set of 0 & 1: {0,1}. I edited accordingly. Also, open-problems was incorrect :-) $\endgroup$ Jan 27, 2014 at 21:57
  • $\begingroup$ Could you clarify what you mean by "triplets of input pulses have no effect on $y$?" $\endgroup$ Jan 28, 2014 at 12:40
  • $\begingroup$ I mean that there are no 3rd order correlations of the input which affect the output. Thus, the effect of a spike triplet can be entirely described in terms of 3 spike pairs & 3 signle spikes. Did I answer you ok? $\endgroup$ Jan 29, 2014 at 22:46

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.