The context for the problem is that I'm working with a modified genetic algorithm where the fitness score of each chromosome is given by a log-probability $ \log(p) \in (-\infty, 0) $. Thus, I have a categorical distribution parameterized by (unnormalized) log-probabilities. These scores are large in magnitude, so I want to be able to sample from this distribution without needing to first exponentiate and normalize, as exponentiating would lead to numerical over/underflow errors during program execution. I know that I can sample proportional to $ p $ without exponentiation by using the Gumbel-max trick.
However, what's giving me trouble is how, given this set of log-probabilities $ \log(p) $ , I would be able to sample from a distribution parameterized by $ \log(1 - p)$ . The motivation for this is that I want to "kill off" lowest-scoring chromosomes in the population, hence driving up overall population fitness. It is not possible to directly convert $\log(p)$ to $\log(1 - p)$ by logarithm rules, and as such I'm struggling to find some kind of transformation that will allow me to do this.
How should I go about thinking of a solution to this? Is this sort of thing even possible?