Compute Scaling Constant for A1 Mapping

Computes the scaling constant \(c_J\) used in the A1 closed-form mapping. Three variants are supported based on asymptotic approximations.

Usage

compute_scaling_constant(
  J,
  scaling = c("log", "harmonic", "digamma"),
  mu_K = NULL
)

Arguments

J

Integer; number of items/sites (must be >= 2).

scaling

Character; one of "log", "harmonic", or "digamma".

mu_K

Numeric; target mean of K (required for "digamma" scaling).

Value

Numeric scalar; the scaling constant \(c_J\).

Details

The scaling constant appears in the Poisson proxy: $$K_J - 1 \mid \alpha \approx \text{Poisson}(\alpha \cdot c_J)$$

Available variants:

log

\(c_J = \log(J)\), the asymptotic leading term (default)

harmonic

\(c_J = H_{J-1} = \psi(J) + \gamma\), improves accuracy for small/moderate J

digamma

\(c_J = \psi(\tilde{\alpha} + J) - \psi(\tilde{\alpha})\) where \(\tilde{\alpha} = (\mu_K - 1)/\log(J)\), a local correction

See also

DPprior_a1 for the main elicitation function

Examples

# Default log scaling
compute_scaling_constant(50, "log")
#> [1] 3.912023

# Harmonic scaling (better for moderate J)
compute_scaling_constant(50, "harmonic")
#> [1] 4.479205

# Digamma scaling (requires mu_K)
compute_scaling_constant(50, "digamma", mu_K = 5)
#> [1] 4.463254