multi_log_score_optimization: Optimize degrees of freedom (nu) via LOO Cross-Validation

In bayesRecon: Probabilistic Reconciliation via Conditioning

Optimize degrees of freedom (nu) via LOO Cross-Validation

Description

Optimize degrees of freedom (nu) via LOO Cross-Validation

Usage

multi_log_score_optimization(res, prior_mean, trim = 0.1)

Arguments

res	Matrix of residuals (n_obs x n_var).
prior_mean	The prior mean covariance matrix (n_var x n_var).
trim	Fraction of observations to trim (0 to 1). Default is 0.1 (10%).

Details

TODO: check these details

Leave-One-Out (LOO) Cross-Validation: This function estimates the optimal degrees of freedom \nu by maximizing the out-of-sample predictive performance. This is achieved by computing the log-density of each held-out observation \mathbf{r}_i given the remaining data \mathbf{R}_{-i}. The total objective function is the sum of these predictive log-densities:

\mathcal{L}(\nu) = \sum_{i=1}^T \log f(\mathbf{r}_i | \mathbf{R}_{-i}, \nu)

The Log-Density Function: For each LOO step, the residuals are assumed to follow a multivariate t-distribution. The density is expressed directly as a function of the posterior sum-of-squares matrix \Psi, where \Psi scales implicitly with \nu:

f(\mathbf{r}_i | \Psi, \nu) = \frac{\Gamma(\frac{\nu + T}{2})}{\Gamma(\frac{\nu + T - p}{2}) \pi^{p/2}} |\Psi|^{-1/2} \left( 1 + \mathbf{r}_i^\top \Psi^{-1} \mathbf{r}_i \right)^{-\frac{\nu + T}{2}}

In the code, \Psi is constructed as:

\Psi = (\nu - p - 1)\bar{\Sigma}_{prior} + \mathbf{R}^\top\mathbf{R}

By using this formulation, the standard scaling factors 1/\nu and \nu^{-p/2} are absorbed into the matrix inverse and determinant, respectively.

Efficient Computation via Sherman-Morrison: Rather than recomputing \Psi_{-i} and its inverse T times, the function uses the full-sample matrix \Psi and adjusts it using the leverage h_i = \mathbf{r}_i^\top \Psi^{-1} \mathbf{r}_i.

Through the Matrix Determinant Lemma and the Sherman-Morrison formula, the internal term (1 + \mathbf{r}_i^\top \Psi_{-i}^{-1} \mathbf{r}_i) simplifies to (1 - h_i)^{-1}. The final log-density contribution used in the code is:

\log f_i \propto \textrm{const} - \frac{1}{2}\log|\Psi| + \frac{\nu + T - 1}{2} \log(1 - h_i)

Value

A list containing the optimization results:

• optimal_nu: The optimal degrees of freedom found.

• min_neg_log_score: The minimum negative log score achieved.

• convergence: The convergence status of the optimizer.

• time_elapsed: The time taken for the optimization.

bayesRecon documentation built on March 8, 2026, 9:08 a.m.