I came across this issue while trying to combine multiple probability distributions into a single distribution which approximates them all simultaneously. This boils down to maximizing this expression
$$
S = \sum\_i \frac{N\_i p\_i^i}{\sum\_j N\_j p\_i^j}
$$
in terms of the unknowns $N\_1, \dots, N\_t$, $p\_1, \dots, p\_t$. Here $p\_i \in [0,1]$ and $N\_i \geq 0$ for all $i$.


It is easy to see that $S \leq t$ (because the denominator term $\sum\_j N\_j p\_i^j \leq N\_i p\_i^i$. Are there any tighter bounds available?


Thanks for the help



Bound on expression from probability distributions
$t$ is in fact a tight bound. It's slightly tricky because the objective is not defined at what should be the optimal solution (due to zeros in numerators and denominators).
What you want is first $p\_1 \to 0+$ (making the first term $ \to N\_1 p\_1/(N\_1 p\_1) = 1$,
then $N\_1 \to 0+$ making the second term $\to N\_2 p\_2^2/(N\_2 p\_2^2 + \ldots)$, then $p\_2
\to 0+$ making the second term $\to 1$, then $N\_2 \to 0+$ etc.


