PMF of K Given Alpha (Antoniak Distribution)

Computes \(P(K_J = k \mid \alpha)\) for \(k = 0, 1, \ldots, J\) using the Antoniak distribution derived from the Dirichlet process.

Usage

pmf_K_given_alpha(J, alpha, logS, normalize = TRUE)

Arguments

J

Integer; sample size (number of observations, must be >= 1).

alpha

Numeric; DP concentration parameter (must be positive scalar).

logS

Matrix; pre-computed log-Stirling matrix from compute_log_stirling.

normalize

Logical; if TRUE (default), apply a final probability-scale normalization. If FALSE, directly exponentiate the already normalized log-PMF.

Value

Numeric vector of length \(J+1\) containing \(P(K_J = k \mid \alpha)\) for \(k = 0, 1, \ldots, J\). Entry [1] corresponds to \(k=0\) and always equals 0. The mathematical support is \(\{1,\ldots,J\}\). The vector sums to 1 up to floating-point tolerance in either normalization mode.

Details

The Antoniak distribution gives the exact PMF of the number of occupied clusters \(K_J\) in a Dirichlet process with concentration parameter \(\alpha\): $$P(K_J = k \mid \alpha) = |s(J,k)| \frac{\alpha^k}{(\alpha)_J}$$

where \(|s(J,k)|\) is the unsigned Stirling number of the first kind and \((\alpha)_J\) is the rising factorial.

Key properties:

• \(P(K_J = 0) = 0\) always (at least one cluster exists)

• \(P(K_J = J) > 0\) for all \(\alpha > 0\)

• As \(\alpha \to 0^+\), mass concentrates on \(K_J = 1\)

• As \(\alpha \to \infty\), mass concentrates on \(K_J = J\)

Extremely small support probabilities may underflow to structural zeros on the probability scale. Setting normalize = TRUE applies a final normalization after exponentiation; normalize = FALSE skips only that final correction and does not return unnormalized Antoniak weights.

References

Antoniak, C. E. (1974). Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems. The Annals of Statistics, 2(6), 1152-1174.

See also

log_pmf_K_given_alpha for log-scale computation, mode_K_given_alpha, cdf_K_given_alpha, quantile_K_given_alpha

Other conditional_K: cdf_K_given_alpha(), cv_K_given_alpha(), dispersion_K_given_alpha(), log_pmf_K_given_alpha(), mean_K_given_alpha(), mode_K_given_alpha(), moments_K_given_alpha(), quantile_K_given_alpha(), var_K_given_alpha()

Examples

# Compute PMF for J=50, alpha=2
logS <- compute_log_stirling(50)
pmf <- pmf_K_given_alpha(50, 2.0, logS)

# Verify normalization
sum(pmf)  # Should be 1
#> [1] 1

# Verify P(K=0) = 0
pmf[1]    # Should be 0
#> [1] 0

# Most likely number of clusters (mode)
which.max(pmf) - 1  # Subtract 1 for 0-indexing
#> [1] 7

# Compare with moments
k_vals <- 0:50
mean_K <- sum(k_vals * pmf)
var_K <- sum(k_vals^2 * pmf) - mean_K^2

# These should match digamma formulas
mean_K_given_alpha(50, 2.0)
#> [1] 7.037626
var_K_given_alpha(50, 2.0)
#> [1] 4.535558