Exact Marginal Moments of K_J under Gamma Prior

Computes the exact marginal mean $E[K_J]$ and variance $Var(K_J)$ when the DP concentration parameter follows a Gamma(a, b) prior.

Usage

exact_K_moments(J, a, b, M = .QUAD_NODES_DEFAULT)

Arguments

J: Integer; sample size (positive integer >= 1).
a: Numeric; shape parameter of Gamma prior (> 0).
b: Numeric; rate parameter of Gamma prior (> 0).
M: Integer; number of quadrature nodes (default: 80).

Value

A named list with components:

mean: Marginal mean $E[K_J | a, b]$
var: Marginal variance $Var(K_J | a, b)$
sd: Marginal standard deviation
cv: Coefficient of variation (sd/mean)

Details

Uses Gauss-Laguerre quadrature to numerically evaluate: $$M_1(a,b) = E_{\alpha \sim \Gamma(a,b)}[\mu_J(\alpha)]$$ $$V(a,b) = E[v_J(\alpha)] + E[\mu_J(\alpha)^2] - M_1^2$$

where $\mu_J(\alpha)$ and $v_J(\alpha)$ are the conditional mean and variance from Module 03.

The Law of Total Variance decomposes the marginal variance into:

Within-alpha variance: $E[v_J(\alpha)]$
Between-alpha variance: $Var(\mu_J(\alpha))$

Key properties:

The mean is bounded: $1 \leq E[K_J] \leq J$
Despite conditional underdispersion ($v_J(\alpha) < \mu_J(\alpha)$), the marginal distribution is typically overdispersed
Marginal variance exceeds conditional variance at $\alpha = E[\alpha]$

References

Antoniak, C. E. (1974). Mixtures of Dirichlet Processes. The Annals of Statistics, 2(6), 1152-1174.

Examples

# Example from RN-01: J=50, Gamma(1.5, 0.5) prior
result <- exact_K_moments(50, 1.5, 0.5)
print(result)
#> $mean
#> [1] 8.355487
#> 
#> $var
#> [1] 22.76895
#> 
#> $sd
#> [1] 4.771682
#> 
#> $cv
#> [1] 0.5710837
#> 

# Compare with conditional variance at E[alpha] = 3
cond_var <- var_K_given_alpha(50, 3.0)
result$var > cond_var  # TRUE: marginal > conditional
#> [1] TRUE

# Verify mean bounds
1 <= result$mean && result$mean <= 50  # TRUE
#> [1] TRUE