Suppose I have two arbitrary discrete probability distributions with the same domain.
I want to convolve the two together to come up with third distribution, however I want them to be weighted.
Here's what I have so far.
Let's say that the distributions' domain goes from $[1,N]$. I get the $N$ probability values of both distributions and perform a DFT. I now have two lists of complex numbers of length $N$.
I now multiply the elements of both lists together $c_i = a_i \times b_i$ and preform a reverse DFT to get the convolved distribution.
This is a normal, unweighted convolution.
Let's say I want "more of distribution $a$ than distribution $b$ inside the convovled distribution $c$". I cannot simply say $c_i = (w_1 \times a_i) \times (w_2 \times b_i)$ where $w_1 > w_2$ and $w_1 + w_2 = 1$ because the individual weights lose their meaning and are applied to both terms.
I'll tell you what I'm doing and why I want this:
I have a list of elements, and I need to select one. I have two very different ways of choosing which element is the "fittest", which absolutely cannot be compared.
I create two probability distribution functions for the list of elements for each of the different fitness metrics and convolve them together. I then choose a random element from the list based on the convolved distribution.
This is fine if I want to give equal weighting to both metrics, but, in fact, I want to arbitrarily specify the weighting of the two metrics when deciding which element to choose.
Any ideas? Can this even be done or am I approaching this from an entirely wrong direction?
If I want to give metric $a$ a weighting of $2$ and metric $b$ a weighting of $1$, would I need to do $c = (a \star(a \star b))$? In which case performing arbitrary weightings would end up being a little cumbersome?