Profile (RE)ML log-likelihood for a single variance ratio

Computes the profile (RE)ML log-likelihood (up to constants) for a given candidate \(\lambda\) and a vector of residual sum-of-squares values \(Q\) across vertices. This is evaluated inside the \(\lambda\) grid search to select the optimal variance ratio per vertex.

Usage

vw_logLL(REML = TRUE, df, Q, logdetV, logdetA)

Arguments

REML

Logical. If TRUE (default), evaluate the REML objective; otherwise the ML objective.

df

Integer. Degrees of freedom: \(N - p\) (REML) or \(N\) (ML).

Q

Numeric vector of length \(C\) (chunk size). Residual sum of squares \(Q = Y^\top V^{-1} Y - \hat\beta^\top X^\top V^{-1} Y\) per vertex. Vertices with Q <= 0 or non-finite values are pre-set to NA by the caller.

logdetV

Numeric scalar. \(\log |V(\lambda)|\), computed as \(\sum_k \log(1 + n_k \lambda)\).

logdetA

Numeric scalar. \(\log |A(\lambda)|\), computed as \(2 \sum_j \log R_{jj}\) from the Cholesky factor.

Value

Numeric vector of length \(C\); the profile log-likelihood value at this \(\lambda\) for each vertex in the chunk. Vertices with NA in Q propagate NA.

Details

REML profile log-likelihood (constants dropped): $$\ell_{\text{REML}} = -\frac{1}{2} \left[ (N-p) \log\!\frac{Q}{N-p} + \log|V| + \log|A| \right]$$

ML profile log-likelihood (constants dropped): $$\ell_{\text{ML}} = -\frac{1}{2} \left[ N \log\!\frac{Q}{N} + \log|V| \right]$$

where \(\log|V|\) and \(\log|A|\) are the pre-computed log-determinants passed as logdetV and logdetA.