The Classical Occupancy Distribution: Computation and Approximation

Abstract

We examine the discrete distributional form that arises from the "classical occupancy problem, "which looks at the behavior of the number of occupied bins when we allocate a given number of balls uniformly at random to a given number of bins. We review the mass function and moments of the classical occupancy distribution and derive exact and asymptotic results for the mean, variance, skewness and kurtosis. We develop an algorithm to compute a cubic array of log-probabilities from the classical occupancy distribution. This algorithm allows the computation of large blocks of values while avoiding underflow problems in computation. Using this algorithm, we compute the classical occupancy distribution for a large block of values of balls and bins, and we measure the accuracy of its asymptotic approximation using the normal distribution. We analyze the accuracy of the normal approximation with respect to the variance, skewness and kurtosis of the distribution. Based on this analysis, we give some practical guidance on the feasibility of computing large blocks of values from the occupancy distribution, and when approximation is required.

Taming strong selection with large sample sizes

“…Given n g , the number of parental lineages n p follows the modified occupancy distribution (also known as the Arfwedson distribution) [Wakeley, 2009, O'Neill, 2019, Johnson et al, 2005:…”

“…The occupancy distribution requires exchangeability of the alleles, which is satisfied by the condition that all parental alleles are derived. Combining the two distributions together through equation 16, we get: We did not find an analytical expression for this sum, but it can be computed efficiently using methods presented in [O'Neill, 2019]. Figure S9A (dotted line) shows the distribution of the number of contributing parental lineages for several selection coefficients with n o = 200.…”

“…The distribution over the number of gametes, n g , is given by the negative binomial, parameterized by the number of successes n o , and the probability of a successful draw is 1 — s . Given n g , the number of parental lineages n p follows the modified occupancy distribution (also known as the Arfwedson distribution) [Wakeley, 2009, O’Neill, 2019, Johnson et al, 2005]: where S 2 ( n g , n p ) is a Stirling number of the second kind, which is the number of ways to partition n g gametes into n p parents (see Johnson et al [2005] section 10.4 for a thorough treatment). The occupancy distribution requires exchangeability of the alleles, which is satisfied by the condition that all parental alleles are derived. Combining the two distributions together through equation 16, we get:

Figure S9: A The distribution of number of required lineages with n 0 = 200.

…”

“…The dotted line indicates regimes where the Gaussian approximation is likely inaccurate. In both panels, N = 1000. We did not find an analytical expression for this sum, but it can be computed efficiently using methods presented in [O’Neill, 2019]. Figure S9A (dotted line) shows the distribution of the number of contributing parental lineages for several selection coefficients with n o = 200.…”

“…Given n g , the number of parental lineages n p follows the modified occupancy distribution (also known as the Arfwedson distribution) [Wakeley, 2009, O’Neill, 2019, Johnson et al, 2005]: where S 2 ( n g , n p ) is a Stirling number of the second kind, which is the number of ways to partition n g gametes into n p parents (see Johnson et al [2005] section 10.4 for a thorough treatment). The occupancy distribution requires exchangeability of the alleles, which is satisfied by the condition that all parental alleles are derived.…”