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Log Marginal PMF of K_J under Gamma Hyperprior

log_pmf_K_marginal.Rd

Computes \(\log P(K_J = k \mid a, b)\) for \(k = 0, 1, \ldots, J\) using log-space mixing for numerical stability.

Usage

log_pmf_K_marginal(
  J,
  a,
  b,
  logS,
  M = .QUAD_NODES_DEFAULT,
  M_verify = NULL,
  abs_tol = 1e-10,
  rel_tol = 1e-08,
  strict = FALSE
)

Arguments

J

Integer; sample size (must be >= 1).

a

Numeric; shape parameter of Gamma prior (must be > 0).

b

Numeric; rate parameter of Gamma prior (must be > 0).

logS

Matrix; pre-computed log-Stirling matrix from compute_log_stirling.

M

Integer; number of quadrature nodes (default: .QUAD_NODES_DEFAULT).

M_verify

Optional independent quadrature order satisfying the package verification rule (at least max(2*M, M+40), within the supported ceiling). When supplied, the selected PMF is compared with a fresh higher-order PMF.

abs_tol, rel_tol

Non-negative absolute and relative tolerances for the selected-versus-verification L1 discrepancy and the PMF-versus-direct- moment checks at both quadrature orders.

strict

Logical; require successful higher-order verification when TRUE; otherwise return the selected PMF with explicit approximation metadata.

Value

Numeric vector of length \(J+1\) containing log-probabilities for \(k = 0, 1, \ldots, J\). Entry [1] corresponds to \(k=0\) and is always -Inf.

Details

This routine normalizes \(P(K_J = \cdot \mid \alpha_m)\) at each quadrature node before mixing, and then mixes in log-space via logsumexp_vec: $$\log p_k \approx \log\sum_m \exp\{\log w_m + \log p_{k\mid m}\}.$$

The log-space computation is essential for numerical stability when:

  • J is large (tail probabilities become very small)

  • Alpha values span a wide range (extreme quadrature nodes)

  • Parameters lead to concentrated distributions

A verified result must pass both the selected-versus-verification PMF L1 budget and an independent cross-representation audit: at each order, the mean and variance represented by the PMF must agree with the direct marginal-moment quadrature. Thus two mutually agreeing but scientifically incorrect PMFs cannot be classified as converged.