Joint density x probability for a mixed continuous/ordinal stacked vector

Given one patient's full stacked mean vector and covariance matrix (across all jointly-fit biomarkers' repeated measurements), computes the exact Gaussian density at the observed values for the continuous-marker rows, times the Gaussian-copula probability that the unobserved ordinal-marker latent scores fall in the box implied by their observed categories' cumulative-link thresholds – conditional on the continuous rows' observed values (via the standard multivariate-normal conditioning/Schur-complement formula). This is the quantity conditionalYT()/conditionalYDT() compute directly as a single Gaussian density when every marker is continuous (see the quadratic-form comment in R/conditionalYT.R); setting oo_idx = integer(0) here reduces to exactly that computation (the full (2*pi)^{-n/2} normalizing constant is kept, not dropped, even though it would cancel in predictRisk()'s own ratio – dynamicPredictionBio() instead divides this function's output by mvtnorm::dmvnorm()-normalized values from conditionalYTBioCopula()/conditionalYDTBioCopula(), so the constant must be correct in absolute terms, not just consistent within one call, for Y_density to remain a genuine, correctly-scaled conditional density – see R/conditionalYTBio.R's analogous all-continuous computation, which keeps its own constant for the same reason). Returns the log of that density/probability, so that it neither overflows nor underflows (see predictRisk() for how it is exponentiated); -Inf if the ordinal box probability is 0.

Usage

mixed_density_prob_copula(
  Sigma_all,
  mu_full,
  y_all_vec,
  alpha_lower_vec,
  alpha_upper_vec,
  cc_idx,
  oo_idx
)

Arguments

Sigma_all

Full stacked covariance matrix for this patient (as returned by build_conditional_design_copula()).

mu_full

Full stacked mean vector, same row order as Sigma_all.

y_all_vec

Full stacked observed-value vector; entries at oo_idx are ignored.

alpha_lower_vec, alpha_upper_vec

Full stacked threshold-bound vectors; entries at cc_idx are ignored.

cc_idx, oo_idx

Integer index vectors (into the stacked row order) of the continuous- and ordinal-marker rows respectively.