Dependence of Boland–Friel–Maire precomputation grids on starting direction

Determine whether the grid-selection procedure proposed by Boland, Friel, and Maire (2018) for precomputing sufficient statistics in Gibbs random fields produces different precomputed grids when initialized with different starting directions for the Hessian-based directional expansion, and, if differences arise, characterize the nature and conditions of this dependence to ensure reproducibility and robustness of the resulting Metropolis–Hastings acceptance ratios.

Background

Precomputation strategies for Gibbs random fields can reduce the cost of evaluating Metropolis–Hastings ratios by building a grid of sufficient-statistic evaluations in parameter space. Boland et al. (2018) propose an approach that constructs this grid around the posterior mode using gradient and Hessian information to choose expansion directions.

The present paper notes a potential ambiguity in that approach: the algorithm’s output may depend on the choice of starting direction used for the directional expansion. Clarifying whether different starting directions lead to different grids, and under what conditions, is important for ensuring deterministic behavior, comparability across studies, and reliable downstream inference.