elbo_functions: ELBO Component Functions

In fda.vi: Functional Data Analysis using Variational Inference

ELBO Component Functions

Description

Internal functions computing the six additive terms of the Evidence Lower Bound (ELBO), which serves as both the convergence criterion and the M-step objective in the VEM algorithm (da Cruz et al., 2024, eq. 36–42).

The per-curve terms (expectedLogLikelihood, elboInclusionTerm, elboBetaTerm, elboThetaTerm) are summed over i=1,\ldots,m. The global terms (elboSigmaPriorTerm, elboTauTerm) contribute once.

elbo_corr assembles all six terms into the total ELBO at fixed w.

elbo_omega wraps elbo_corr as a function of w alone, passed to optim() as the M-step objective.

dev_elbo provides the analytic gradient \partial\text{ELBO}/\partial w, passed to optim() to enable L-BFGS-B optimisation.

Arguments

y	List of observed curve vectors.
ni	Integer vector of per-curve sample sizes.
B	List of basis matrices.
i	Integer. Curve index (per-curve terms only).
m	Integer. Number of curves.
K	Integer. Number of basis functions.
iter	Integer. Current iteration index.
delta_1, delta_2	Prior hyperparameters for \sigma^2.
lambda_1, lambda_2	Prior hyperparameters for \tau^2.
delta1_q, delta2_values	Shape and scale trajectory of q(\sigma^2).
lambda1_q, lambda2_values	Shape and scale trajectory of q(\tau^2).
mu_beta_values	Matrix of posterior beta mean trajectories.
Sigma_beta	Array of posterior beta covariance matrices.
a1_values, a2_values	Shape parameter trajectories for q(\theta_{ki}).
prob_values	Matrix of inclusion probability trajectories.
mu_ki	Scalar prior hyperparameter for \theta_{ki}.
psi	Current correlation matrix \Psi(w).
w	Positive scalar. Decay parameter (elbo_omega, dev_elbo only).
Xt	Numeric vector of evaluation points (elbo_omega, dev_elbo only).
logdet_psi	Pre-computed \log|\Psi| or NULL (expectedLogLikelihood only).

Value

A numeric scalar.

fda.vi documentation built on June 20, 2026, 5:06 p.m.