A1 Closed-Form Prior Elicitation

Maps target beliefs about the number of clusters $(\mu_K, \sigma^2_K)$ to Gamma hyperprior parameters $(a, b)$ using the A1 closed-form approximation based on Negative Binomial moment matching.

Usage

DPprior_a1(
  J,
  mu_K,
  var_K,
  scaling = c("log", "harmonic", "digamma"),
  epsilon = .TOL_PROJECTION_BUFFER
)

Arguments

J: Integer; number of items/sites (must be >= 2).
mu_K: Numeric; target prior mean of $K_J$ (must be > 1 and <= J).
var_K: Numeric; target prior variance of $K_J$ (must be > 0).
scaling: Character; scaling constant method: "log" (default), "harmonic", or "digamma".
epsilon: Numeric; buffer for feasibility projection. Default is .TOL_PROJECTION_BUFFER (1e-6).

Value

An S3 object of class DPprior_fit with components:

a: Shape parameter of the Gamma prior
b: Rate parameter of the Gamma prior
J: Sample size used
target: List with target moments and type
method: "A1" indicating closed-form method
status: "success" or "projected" if boundary adjustment needed
scaling: Scaling method used
cJ: Scaling constant value
var_K_used: Actual variance used (may differ if projected)
converged: Always TRUE for A1 (for A2 compatibility)
iterations: Always 0L for A1 (for A2 compatibility)
fit: NULL (placeholder for A2 refinement)
diagnostics: NULL (placeholder for diagnostics)
trace: NULL (placeholder for optimization trace)

Details

Theory (RN-03)

The A1 method uses a shifted Negative Binomial approximation: $$K_J - 1 \mid \alpha \approx \text{Poisson}(\alpha \cdot c_J)$$

With $\alpha \sim \text{Gamma}(a, b)$, the marginal becomes: $$K_J - 1 \approx \text{NegBin}(a, b/(b + c_J))$$

Inverse Formulas (Theorem 1)

Let $m = \mu_K - 1$ (shifted mean) and $D = \sigma^2_K - m$. If $D > 0$ (overdispersion): $$a = m^2 / D, \quad b = m \cdot c_J / D$$

If $D \leq 0$ (infeasible), the variance is projected to the boundary.

Feasibility

The NegBin model requires overdispersion: $\sigma^2_K > \mu_K - 1$. High-confidence specifications (low variance) may violate this constraint under the A1 proxy, even though they may be feasible under the exact DP.

Examples

# Basic usage with moment targets
fit <- DPprior_a1(J = 50, mu_K = 5, var_K = 8)
print(fit)
#> DPprior Prior Elicitation Result
#> ============================================= 
#> 
#> Gamma Hyperprior: α ~ Gamma(a = 4.0000, b = 3.9120)
#>   E[α] = 1.022, SD[α] = 0.511
#> 
#> Target (J = 50):
#>   E[K_J]   = 5.00
#>   Var(K_J) = 8.00
#> 
#> Method: A1 (0 iterations)

# Using VIF specification
fit <- DPprior_a1(J = 50, mu_K = 5, var_K = vif_to_variance(5, 2))

# Using confidence-based specification
vif <- confidence_to_vif("medium")
fit <- DPprior_a1(J = 50, mu_K = 5, var_K = vif_to_variance(5, vif))

# Infeasible variance (triggers projection)
fit <- DPprior_a1(J = 50, mu_K = 5, var_K = 3)  # var < mu-1 = 4
#> Warning: var_K <= mu_K - 1: projected to feasible boundary

# Compare scaling methods
fit_log <- DPprior_a1(50, 5, 8, scaling = "log")
fit_harm <- DPprior_a1(50, 5, 8, scaling = "harmonic")