Conditional Mean of K_J Given Alpha

Computes \(\mathbb{E}[K_J \mid \alpha]\) under a Dirichlet process prior.

Usage

mean_K_given_alpha(J, alpha)

Arguments

J

Sample size (integer >= 1).

alpha

Concentration parameter (positive numeric, vectorized).

Value

Numeric vector of conditional means (same length as alpha).

Details

Uses the digamma closed form: $$\mu_J(\alpha) = \alpha\{\psi(\alpha+J)-\psi(\alpha)\}$$ where \(\psi(\cdot)\) is the digamma function.

This is equivalent to the direct summation: $$\mu_J(\alpha) = \sum_{i=1}^{J} \frac{\alpha}{\alpha + i - 1}$$

Limiting behavior:

• \(\alpha \to 0^+\): \(\mu_J(\alpha) \to 1\)

• \(\alpha \to \infty\): \(\mu_J(\alpha) \to J\)

For numerical stability, values with alpha < 1e-10 return the limit 1.

See also

var_K_given_alpha, moments_K_given_alpha

Examples

mean_K_given_alpha(50, 2.0)
#> [1] 7.037626
mean_K_given_alpha(50, c(0.5, 1, 2, 5))
#> [1]  2.937775  4.499205  7.037626 12.460485

# Limiting behavior
mean_K_given_alpha(50, 1e-10)  # Returns 1
#> [1] 1
mean_K_given_alpha(50, 1e6)    # Returns ~50
#> [1] 49.99878