A simple and fast method for computing the Poisson binomial distribution function

It is shown that the Poisson binomial distribution function can be efficiently calculated using simple convolution based methods. The Poisson binomial distribution describes how the sum of independent but not identically distributed Bernoulli random variables is distributed. Due to the intractability of the Poisson binomial distribution function, efficient methods for computing it have been of particular interest in past Statistical literature. First, it is demonstrated that simply and directly using the definition of the distribution function of a sum of random variables can calculate the Poisson binomial distribution function efficiently. A modified, tree structured Fourier transform convolution scheme is then presented, which provides even greater gains in efficiency. Both approaches are shown to outperform the current state of the art methods in terms of accuracy and speed. The methods are then evaluated on a real data image processing example in order to demonstrate the efficiency advantages of the proposed methods in practical cases. Finally, possible extensions for using convolution based methods to calculate other distribution functions are discussed.