Computes the PMF of the Beta-Binomial distribution parameterised by mean \(\mu \in [0, 1]\) and concentration \(\kappa > 0\). The shape parameters are \(a = \mu \kappa\) and \(b = (1 - \mu) \kappa\).
Details
The PMF is $$ f(y \mid n, \mu, \kappa) = \binom{n}{y} \frac{B(y + a,\; n - y + b)}{B(a, b)}, \quad y \in \{0, 1, \ldots, n\}, $$ where \(a = \mu \kappa\) and \(b = (1 - \mu) \kappa\).
Boundary handling. When \(\mu = 0\) the Beta-Binomial degenerates
to a point mass at \(y = 0\) (all probability on zero). When
\(\mu = 1\) it degenerates to a point mass at \(y = n\). These
boundaries are handled with explicit branches because the lbeta
formula produces NaN when \(a = 0\) or \(b = 0\).
All arguments are recycled to common length via rep_len.
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.
See also
Other distributions:
compute_p0(),
compute_ztbb_mean(),
dhurdle_betabinom(),
dztbetabinom(),
hurdle_mean(),
hurdle_variance(),
pbetabinom(),
rbetabinom(),
rhurdle_betabinom(),
rztbetabinom()
Examples
# Standard usage
dbetabinom(0:5, n = 5, mu = 0.3, kappa = 10)
#> [1] 0.23076923 0.31468531 0.25174825 0.13986014 0.05244755 0.01048951
dbetabinom(0:5, n = 5, mu = 0.3, kappa = 10, log = TRUE)
#> [1] -1.466337 -1.156182 -1.379326 -1.967112 -2.947942 -4.557380
# Boundary: mu = 0 gives point mass at y = 0
dbetabinom(0:3, n = 3, mu = 0, kappa = 5)
#> [1] 1 0 0 0
# Boundary: mu = 1 gives point mass at y = n
dbetabinom(0:3, n = 3, mu = 1, kappa = 5)
#> [1] 0 0 0 1