Score a single feature given responsibilities (Gamma)

Computes an alignment score for a single feature \(x_j\) under a fixed responsibilities matrix \(\Gamma \in \mathbb{R}^{n\times K}\). The score is used for greedy feature partitioning into multiple latent orderings.

Two score modes are supported:

score_mode = "none"

Plug-in (ELBO/Q-like) score. The smoothing parameter \(\lambda\) is fixed at lambda_init (default 1). The score depends on the MAP component means \(\hat\mu_{j\cdot}\) via the weighted SSE.

score_mode = "ml"

Collapsed (marginal-like) score: plug-in score plus the Laplace curvature correction \(+\frac{K}{2}\log(2\pi) - \frac12\log|A_j|\). In this mode, \(\lambda\) is optimized by 1D maximization over \(\log \lambda \in [\log(\text{lambda_min}),\log(\text{lambda_max})]\).

The fitted 1D quantities \(\hat\mu_{j\cdot}\) and \(\hat\sigma_j^2\) are returned and can be appended to a partition's csmooth-style parameters during greedy growth.

score_feature_given_Gamma(
  xj,
  Gamma,
  Q_K,
  rw_q = 0L,
  score_mode = c("ml", "none"),
  relative_lambda = TRUE,
  lambda_min = 1e-10,
  lambda_max = 1e+10,
  nugget = 0,
  optimize_lambda = NULL,
  lambda_init = 1,
  max_sigma_iter = 1L
)

Arguments

xj

Numeric vector of length n (one feature).

Gamma

Numeric matrix (n x K) of responsibilities.

Q_K

Numeric matrix (K x K); base RW precision (lambda=1).

rw_q

Integer \(\ge 0\). Rank deficiency along K (RW order).

score_mode

One of "ml" or "none".

relative_lambda

Logical; if TRUE use \(Q_{base} = Q_K / \sigma_j^2\) (homoskedastic scaling).

lambda_min, lambda_max

Positive bounds for \(\lambda\) when score_mode="ml".

nugget

Nonnegative scalar added to \(\sigma_j^2\).

optimize_lambda

Logical or NULL. If NULL, defaults to TRUE for "ml" and FALSE for "none".

lambda_init

Positive scalar. Fixed \(\lambda\) when score_mode="none" (default 1), and initial value if optimize_lambda=FALSE.

max_sigma_iter

Integer \(\ge 1\). Number of (lambda -> mu -> sigma2) refresh steps.

Value

A list with components:

  • score: scalar score for feature \(j\).

  • lambda: fitted (or fixed) \(\lambda_j\).

  • sigma2: fitted \(\sigma_j^2\).

  • mu_vec: numeric vector length K of \(\hat\mu_{j\cdot}\).

  • logdetA: scalar \(\log|A_j|\) (only meaningful for "ml"; still returned).