Systematic study of k-wise and hypergraph convergence diagnostics

Investigate k-wise extensions of the MCMC convergence graph based on k-wise convergence statistics and develop corresponding hypergraph diagnostics, including their properties and practical utility.

Background

The paper’s MCMC convergence graph uses pairwise \hat{R}_{ij} statistics between two chains, producing an ordinary graph whose connected components summarize groups of chains that explore the same posterior region. The authors note that standard convergence diagnostics generally recommend at least four chains, motivating a possible generalization from pairwise (k=2) statistics to statistics computed over k chains.

The proposed k-wise statistics are intended to satisfy analogues of symmetry, agreement, and disagreement: they should be invariant to permutations of the chain indices, converge to 1 when all k chains target the same restricted distribution, and converge to a value greater than 1 otherwise. The authors retain the pairwise construction because it is sufficient for partition recovery in the asymptotic transitive-agreement regime and yields an ordinary graph that can be directly visualized. They explicitly leave a systematic investigation of higher-order statistics and hypergraph-based diagnostics unresolved.