# Poisson Binomial Distribution - confidence intervals

I'm working on a project which involves multiple trials for which the probability of success is not the same across trials. Given the unequal probabilities per trial, I'm using the Poisson Binomial distribution.

I can easily model the probability of a given number of successes across all trials using the R package poisbinom. Looking at the problem the other way around, I would also like to provide confidence intervals on the probability of successes given an observed number of successes.

For the Binomial Distribution, I know this can be done using a normal approximation when $$p>0.1$$ and $$n>15$$, but I cannot find a reference for computing CIs for the Poisson Binomial Distribution. This is obviously complicated by the fact that each trial has a different probability of success

Any references to using a normal approximation or other methods is appreciated.

EDIT:

I've been doing some simulations to see if a normal approximation (Wald Method) would work, and generally I see the same results I would expect to see as with a Binomial Distribution. The empirical variability I see from the simulation is certainly non-normal, but normal approximation does a reasonably good job with 2 decimal points - 0.## +/- 0.## when the sample size is large.

From some background reading, the same deviation from normal is seen for the Binomial Distribution and various approximation methods are proposed. The Wald Method is simple but biased.

From what I've seen from simulations, I think similar CI methods as those used for a Binomial Distribution would work.

UPDATE:

I've tried multiple types of CI calculation that are applicable to the Binomial Distribution, or more specifically, the Binomial Proportion - Wald, Agresti-Coull, Wilson. see here for descriptions From my simulations, Wald performs best across the broadest range of values. The others are generally more often recommended, but I think that is primarily for proportions that are very small (< 0.01) or very large (> 0.99).

I feel I've answered my own question here, but I appreciate any feedback.

• If you only have 1 sequence of the n trials & only know the success parameters are unequal without knowing a given relationship, I don't see how you can estimate the $p_i$s much less get confidence intervals. It seems that the most you can say is that for the $p_i$s that have a failure are <1. – Michael R. Chernick Nov 4 '19 at 15:04

It is not clear what you really want. The Poisson-Binomial distribution on $$n$$ trials have $$n$$ probability parameters $$p_1, \dotsc, p_n$$ so you cannot reasonably expect a confidence interval for each of them! So, for this question to be answerable, you need to specify some function of the $$p_i$$'s that you are interested in. So, what is your real inferential goal? Let's say your Poisson-Binomial random variable is $$X=\sum_{i=1}^n X_i$$ where the $$X_i$$'s are independent Bernoulli with probability $$p_i$$. Maybe your interest is in

1. $$\DeclareMathOperator{\P}{\mathbb{P}} \bar{p}=\frac1{n} \sum p_i$$

2. The fraction of the $$p_i$$'s above 0.7?

3. The variance of the $$p_i$$'s?

4. $$\P(X \ge 0.78 n)$$?

or some other functional? As soon as you specify your focus parameter, or otherwise your inferential goal, we can probably propose some confidence interval.

Your question also could benefit from some context. The Poisson-Binomial distribution is so general, more of a modeling framework, so context could give important hints to modeling. Some papers leading to ideas: Probability of forest fires, Species Distribution Models, Species Distribution models 2, simulation-based CI (JSTOR).