Expected number of clusters under the Gnedin prior

Compute the closed-form mean of \(K_n\) (number of clusters) under the Gnedin finite-type model (\(\sigma=-1\)).

Usage

gnedin_K_mean(n, gamma)

Arguments

n

Integer vector of sample sizes (each \(\ge 1\)).

gamma

Numeric vector of Gnedin parameters, each in \((0,1)\).

Value

A numeric vector with \(\mathbb{E}[K_n]\) for each pair (n, gamma).

Details

For sample size \(n \ge 1\) and parameter \(\gamma \in (0,1)\), the mean is $$ \mathbb{E}[K_n] \;=\; \frac{\Gamma(n+1)\,\Gamma(1+\gamma)}{\Gamma(n+\gamma)}. $$ The implementation uses lgamma for numerical stability and is vectorized over n and gamma (with standard R recycling rules).

The formula follows directly from standard Gibbs–type manipulations using factorial moments and the Chu–Vandermonde identity specialized to \(\sigma=-1\).

References

Gnedin, A. (2010). A species sampling model with finitely many types. Electronic Communications in Probability, 15, 79–88.

Favaro, S., Lijoi, A., & Prünster, I. (2013). Extending the class of Gibbs-type priors: theoretical properties and new examples. Annals of Applied Probability, 23(4), 1729–1754.

Pitman, J. (2006). Combinatorial Stochastic Processes. Springer.

See also

gnedin_K_var()

Examples

# Scalar inputs
gnedin_K_mean(105, 0.8)
#> [1] 2.36427

# Vectorized over gamma
gnedin_K_mean(50, c(0.3, 0.5, 0.8))
#> [1] 13.906329  6.282256  2.039934

# Vectorized over n and gamma (recycling rules)
gnedin_K_mean(c(20, 50, 100), 0.5)
#> [1] 3.988173 6.282256 8.873354