Characterize the tightest posterior-attraction set

Characterize in general the tightest set to which the Bayesian posterior concentrates, beyond the bounds established in Supplemental Appendix Section S.7, by resolving how log-likelihood fluctuations compete with multiplicity-induced logarithmic factors.

Background

The paper defines the tightest set as the intersection of all closed sets that attract the posterior and proves that it lies between the concurrence set and the concentration set. The authors do not obtain a general characterization.

They explain that membership depends on the competition between stochastic fluctuations of the log-likelihood and logarithmic multiplicity factors, and propose a conjectural route involving the cluster set of a normalized empirical process.

References

Lastly, Supplemental Appendix \ref{supp:tightest} brackets the tightest set to which the posterior concentrates, but we were unable to characterize it in general. Whether a parameter belongs to it depends on how the fluctuations of the log-likelihood compete with the logarithmic factors from multiplicity, and we conjecture that a characterization requires the cluster set of the normalized empirical process, which, by the compact law of the iterated logarithm, is the unit ball of the reproducing kernel Hilbert space of its covariance.

— Singular Equilibrium and Selection of Narratives  (2610.11615 - Jin, 8 Oct 2026) in Section 6, Concluding Remarks; Supplemental Appendix, Section S.7 (Tightest Convergence)