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

Multilevel Monte Carlo (MLMC) methods for Bayesian inverse problems are now mature,with rigorous complexity analyses established under standard regularity assumptions 22, 19.However, two fundamental gaps remain. First, recent work by O’Brien et al. 37 revealsthat numerical discretization error can induce severe posterior pathologies—including spuriousbimodality—that are neither predicted nor prevented by existing multilevel theory. Second, thepush toward robust Bayesian inference with heavy-tailed priors 38 has proceeded independentlyof discretization error analysis; it remains unknown how numerical approximation interacts withnon-Gaussian prior tails.This paper provides the first unified analysis of telescoping Bayesian inference undernon-ideal conditions. Our contributions are threefold:1. Numerical pathology theory: We prove that spurious bimodality arises precisely whentelescoping increments are oscillatory (e.g., Runge–Kutta 4) rather than sign-definite (e.g.,spectral methods). This provides the first mathematical explanation of the O’Brien et al.phenomenon and yields a sharp threshold: methods with order k ≥ 2 and sign-definite errorguarantee eventual unimodality.2. Heavy-tail penalty: We derive the exact convergence rate degradation for priors withpolynomial tails ∼ ∥x∥−α. The rate slows from O(n−(k+1)) to O(n−α−2α−1 (k+1)), quantifyingthe cost of robustness. This bridges the gap between the heavy-tailed prior literature andnumerical analysis.3. Adaptive stopping: We introduce the first posterior-driven adaptive refinement algorithmthat achieves optimal O(ε−2) complexity without requiring a priori knowledge of the convergencerate k or cost exponent β. The stopping rule is derived directly from the telescopingvariance decomposition.Numerical experiments confirm the theoretical rates and demonstrate that the adaptiveestimator matches oracle performance with only 12–15% overhead. Our framework does not replaceexisting multilevel methods but rather provides the analytic tools to diagnose pathologies,quantify robustness costs, and automate refinement.