Log-PMF of K Given Alpha (Antoniak Distribution)
log_pmf_K_given_alpha.RdComputes \(\log P(K_J = k \mid \alpha)\) for \(k = 0, 1, \ldots, J\).
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.
Value
Numeric vector of length \(J+1\) containing
\(\log P(K_J = k \mid \alpha)\) for \(k = 0, 1, \ldots, J\).
Entry [1] corresponds to \(k=0\) and always equals -Inf.
The mathematical support is \(\{1,\ldots,J\}\); the leading structural
zero is retained in the vector representation for compatibility.
Details
Uses the Antoniak distribution formula in log-space: $$\log P(K_J = k \mid \alpha) = \log|s(J,k)| + k\log\alpha - \log(\alpha)_J$$
where \(|s(J,k)|\) is the unsigned Stirling number of the first kind and \((\alpha)_J\) is the rising factorial. The implementation normalizes the equivalent log-weights over \(k=1,\ldots,J\) with log-sum-exp. This avoids cancellation in a separately evaluated log-gamma normalizer at very large finite \(\alpha\).
This log-space computation is numerically stable for large \(J\) where direct computation would overflow.
See also
pmf_K_given_alpha for normalized PMF,
compute_log_stirling for Stirling computation
Other conditional_K:
cdf_K_given_alpha(),
cv_K_given_alpha(),
dispersion_K_given_alpha(),
mean_K_given_alpha(),
mode_K_given_alpha(),
moments_K_given_alpha(),
pmf_K_given_alpha(),
quantile_K_given_alpha(),
var_K_given_alpha()
Examples
logS <- compute_log_stirling(50)
log_pmf <- log_pmf_K_given_alpha(50, 2.0, logS)
# Convert to probabilities (numerically stable softmax)
pmf <- exp(log_pmf - max(log_pmf))
pmf <- pmf / sum(pmf)
sum(pmf) # Should be 1
#> [1] 1
# Or use pmf_K_given_alpha() directly
pmf2 <- pmf_K_given_alpha(50, 2.0, logS)
sum(pmf2) # Should be 1
#> [1] 1