# Poisson Binomial Distribution - confidence intervals for sum of unequal probabilities estimated with uncertainty

I believe my question is related to, but distinct from this one: Poisson Binomial Distribution - confidence intervals

I am working to estimate species richness by summing the results of 12 individual species distribution models (Maxent) and then determine whether the richness falls above or below certain thresholds.

So, for each species I estimate a probability of occurrence and I can then sum the probabilities to estimate the richness. The probabilities are unequal so the species richness estimate should follow a Poisson binomial distribution, and as discussed in this paper, estimating the variance of the species richness estiamte would be straightforward if the probabilities were known exactly: http://portal.uni-freiburg.de/biometrie/mitarbeiter/dormann/calabrese2013globalecolbiogeogr.pdf

Var(Richness)= sum((1-ps)*ps)

where ps is a vector of the probabilities of occurrence for the 12 individual species.

However, each probability is estimated with uncertainty. I've used 30 fold cross-validation to get an estimate of the uncertainty. I'd now like to construct a confidence interval (or something similar)

I did have one idea, but I'm not sure if it's valid:

Because I used cross-validation, I have a measure of uncertainty for each of the 12 values. My thought was that I could use bootstrapping to come up with a confidence interval (or something similar) for the sum as follows:

1. Randomly select the prediction from 1 of the 30 model-replicates (from the cross validation) for each of the 12 values.
2. Sum the 12 values.
3. Repeat 1000 times.
4. Take the 2.5% and 97.5% quantile from the thousand replicates as the lower and upper bounds.

Is this appropriate? Is there a better way to use the information I have to do this?

• Could this be a bit simpler - just do 1000-fold cross-validation, summing the 12 values each time, and then taking the 2.5% and 97.5% values? I don't see the advantage of the 30-fold step. – rw2 Dec 16 '19 at 12:08
• Something simpler might be possible. This might be simpler but it has a major downside. The 30-fold cross validation is already done, so I've been trying to work with that- it uses each observation only once to estiamte things. I could go to a leave one out cross validation or something else that reuses the data in fitting the models for the individual species but even with a rather powerful machine, the computation time for the kind of model/software I'm using would be would be very long- MUCH MUCH longer than what I was asking about....I'm not sure its really feasible....I could look into it – user2870897 Dec 16 '19 at 19:38