Bayesian Telescoping Under Non-Ideal Conditions: Numerical Pathologies, Heavy-Tailed Priors, and Adaptive Stopping

Multilevel and telescoping methods for Bayesian inverse problems are well developed under idealconditions. This paper is a corrected revision of an earlier manuscript that claimed threeresults beyond the standard theory. Two of the three claims do not survive scrutiny; the third isprovable only in a restricted form, which we give. Specifically, (i) the multilevel regimethresholds are γ=β+2, not γ=β+1, and the sub-optimal exponent is-2-(β+2-γ)/k (the bias is O(L⁻ᵏ), so Lε-1/k); (ii) discretization-induced bimodality cannot arise from alinear forward operator with a Gaussian prior --- the approximate posterior is then exactly Gaussianat every level --- and for the Runge--Kutta~4 surrogate of y=λ y it is acoarse-resolution transient caused by non-injectivity of the numerical map, which disappearsonce the fold λ=z^ n/T leaves the prior's effective support, rather than persisting atall levels; and(iii) the claimed heavy-tail penalty ρ(α)=(α-2)/(α-1) is neither proved norsupported by the earlier paper's own experiment, which we reproduce and which in fact shows the fullrate $-(k+1)$ for every tail index tested. We state precisely what is proved, what is verifiednumerically, and what remains open, and we ship a verification driver that regenerates every numeralin this paper.