Compute P(Y = 0) under the Beta-Binomial distribution

Computes the probability of observing zero under a $\text{BetaBin}(n, \mu, \kappa)$ distribution using the stable lgamma formula.

Usage

compute_p0(n, mu, kappa)

Arguments

n: Integer vector of trial sizes ($n \ge 0$).
mu: Numeric vector of means, each in $[0, 1]$.
kappa: Numeric vector of concentrations, each $> 0$.

Value

A numeric vector of probabilities in $[0, 1]$.

Details

The formula is $$ p_0 = \frac{B(b + n,\; a)}{B(b,\; a)} = \exp\bigl[\Gamma(b + n) + \ln\Gamma(\kappa) - \ln\Gamma(b) - \ln\Gamma(\kappa + n)\bigr], $$ where $a = \mu\kappa$ and $b = (1 - \mu)\kappa$.

Boundary handling:

$\mu = 0$: Returns 1 (point mass at zero).
$\mu = 1$: Returns 0 (point mass at $n$).
$n = 0$: Returns 1 (trivially, no trials).

The result is clamped to $[0, 1]$ to guard against floating-point drift.

References

Ghosal, S., Ghosh, S., and Moores, M. (2020). “Hierarchical beta-binomial models for batch effects in cytometry data.” Journal of the Royal Statistical Society: Series A, 183(4), 1579–1601.

Examples

# Small n: compare with dbetabinom
compute_p0(5, mu = 0.3, kappa = 10)
#> [1] 0.2307692
dbetabinom(0, n = 5, mu = 0.3, kappa = 10)
#> [1] 0.2307692

# Boundary cases
compute_p0(5, mu = 0, kappa = 10)    # 1
#> [1] 1
compute_p0(5, mu = 1, kappa = 10)    # 0
#> [1] 0
compute_p0(0, mu = 0.3, kappa = 10)  # 1
#> [1] 1