Bayesian Telescoping: Hierarchical Inference and Uncertainty Quantification

We introduce a Bayesian formulation of telescoping approximations, extending the frameworkdeveloped for deterministic, stochastic, data-driven, and inverse problems to hierarchicalprobabilistic inference. Rather than treating discretization or model error as a nuisance, weencode telescoping refinement levels directly into the Bayesian hierarchy.We show that telescoping forward approximations induce a natural multilevel posterior structurein which successive refinements contribute diminishing information at an explicitly quantifiablerate. Under standard assumptions on the forward operator and prior measure, posteriorexpectations, variances, and credible sets inherit telescoping convergence with explicit rates.This provides principled uncertainty quantification that accounts jointly for noise, regularization,and model refinement.The framework unifies several distinct methodologies: it recovers multilevel Monte Carlo(MLMC) as a special case, provides rigorous justification for hierarchical approximations ininverse problems, and offers computable error bounds for posterior quantities. We prove thatif the forward model admits an order-k telescoping approximation, then posterior expectationsconverge at rate O(n−(k+1)) and posterior variances decay at the same rate.We provide an increment-wise perturbation analysis of Bayesian posteriors induced by deterministictelescoping forward-model refinements, yielding explicit level-to-level bounds thatcan be paired with existing multilevel estimators and samplers.Applications include Bayesian inverse problems for partial differential equations, uncertaintyawaresurrogate models, sequential data assimilation, and multilevel Monte Carlo from a deterministictelescoping viewpoint. Numerical experiments on heat equation parameter inference,diffusion coefficient estimation, and nonlinear inverse problems demonstrate the theoretical convergencerates and computational efficiency of the approach.