Tighten the precision-grid size bound

Determine the extent to which the number of logarithmically spaced precision-grid points Q required by the thermodynamic-integration posterior sampler can be tightened, thereby improving the sampler's complexity dependence on the target accuracy.

Background

The sampler discretizes the unknown noise precision parameter $\tau$ onto a logarithmically spaced grid with $Q$ points and estimates the resulting categorical posterior using thermodynamic integration. The paper derives polynomial complexity bounds for total-variation and Wasserstein-1 accuracy. In the discussion following the Wasserstein-1 corollary, the authors identify the $\eta^4$ dependence in the complexity bound as arising entirely from the choice of $Q$. The unresolved issue is therefore to determine whether the required grid size, or its contribution to the accuracy guarantee, can be reduced without invalidating the non-asymptotic posterior-sampling bound.