I am trying to derive a nice expression (without going through Taylor Expansion) of the following formula:
$$ \prod_{j=1}^{k} (1-1/2\exp(x_j)),$$
Splitting it you can see that for $k=3$, for instance,
$$ 1-1/2\sum_je^{x_j} + 1/(2^2) \sum_{i\neq j}^{k}e^{(x_i+x_j)}-1/(2^3)e^{(x_1+x_2+x_3)}.$$
I am trying to get an expression for general $k$ but I am really struggling, do you have any idea on how to do this?
Another thing I could do is upper-bound every $-\exp(x_i)$ with an $-\exp(x)$ (with $x$ constant) and replace the formula with:
$$(1-1/2\exp(x))^k$$
Is there a nice way to express this as a sum?
Best Answer
The expression you want is: $$\prod_{j=1}^k\left(1-\frac{1}{2}\exp(x_j)\right) = \sum_{S\subseteq\{1,\ldots,k\}}\left(-\frac{1}{2}\right)^{|S|}\exp\left(\sum_{j\in S}x_j\right).$$ I leave it to you to check that it is correct (hint: induction).