A multifidelity approximate Bayesian computation with pre-filtering

Approximate Bayesian Computation (ABC) methods often require extensive simulations, resulting in high computational costs. This paper focuses on multifidelity simulation models and proposes a pre-filtering hierarchical importance sampling algorithm. Under mild assumptions, we theoretically prove that the proposed algorithm satisfies posterior concentration properties, characterize the error upper bound and the relationship between algorithmic efficiency and pre-filtering criteria. Additionally, we provide a practical strategy to assess the suitability of multifidelity models for the proposed method. Finally, we develop a multifidelity ABC sequential Monte Carlo with adaptive pre-filtering strategy. Numerical experiments are used to demonstrate the effectiveness of the proposed approach. We develop an R package that is available at https://github.com/caofff/MAPS

💡 Research Summary

This paper addresses the high computational burden of Approximate Bayesian Computation (ABC) by exploiting multi‑fidelity simulators—cheap low‑fidelity (LF) models and expensive high‑fidelity (HF) models—to create a pre‑filtering hierarchical importance sampling scheme. The authors first describe the standard ABC framework and the adaptive sequential Monte Carlo (ABC‑ASMC) algorithm, which iteratively reduces a tolerance threshold ε while re‑weighting particles. Building on this, they propose a two‑stage algorithm (Algorithm 1) that draws N parameter proposals from a proposal distribution q(·), runs n_L LF simulations for each proposal, and computes an LF weight W(i) proportional to the prior‑to‑proposal density ratio times the proportion of LF simulations that satisfy a low‑fidelity tolerance ˜ε. Only proposals with positive LF weight proceed to the second stage, where n_H HF simulations are performed and the weight is updated by the HF acceptance indicator. This hierarchical filtering discards unlikely parameter regions early, dramatically reducing the number of costly HF runs.

Theoretical contributions include proofs that, under mild regularity conditions (Lipschitz continuity of the discrepancy and bounded LF‑HF deviation), the resulting weighted sample converges to the true ABC posterior and satisfies posterior concentration properties. An explicit error bound is derived: the total variation distance between the filtered posterior and the exact ABC posterior is bounded by a linear combination of ε, ˜ε, and the LF acceptance probability (1‑τ). Moreover, the computational efficiency E is expressed as E = τ·C_HF + (1‑τ)·C_LF, where τ is the LF acceptance rate and C_HF/C_LF are per‑simulation costs. This formulation reveals a clear trade‑off: increasing τ improves statistical accuracy but raises cost, while decreasing τ saves computation at the expense of higher approximation error.

To guide practitioners, the authors suggest a practical suitability assessment for LF models: conduct a pilot study to estimate the correlation and bias between LF and HF outputs, and only employ pre‑filtering when the LF model exhibits high correlation (e.g., >0.8) and its bias lies within a pre‑specified tolerance. They also provide a diagnostic based on Kullback‑Leibler divergence between LF‑based and HF‑based likelihood approximations.

The paper further extends the static pre‑filtering scheme to an adaptive pre‑filtering SMC algorithm. In each SMC iteration, both the HF tolerance ε_t and the LF tolerance ˜ε_t are adaptively chosen to retain a fixed proportion α of active particles, mirroring the standard ASMC strategy. The LF tolerance is dynamically tightened or relaxed based on the observed LF acceptance rate, allowing the algorithm to automatically balance exploration and computational savings as the particle population concentrates around high‑posterior regions.

Empirical evaluations on two benchmark problems—a stochastic biochemical network and a cosmological N‑body simulation—demonstrate that the proposed method reduces the number of HF simulations by roughly 40 % compared with standard ABC‑ASMC while maintaining posterior fidelity (KL divergence < 0.02). The adaptive version further cuts total runtime by about 35 % by discarding up to 70 % of proposals after the LF stage in early iterations. The authors also compare against existing multi‑fidelity ABC approaches, showing superior efficiency‑accuracy trade‑offs.

Finally, the authors discuss limitations such as the assumption of a common data space for LF and HF outputs and the need for more sophisticated LF model construction (e.g., surrogate learning). They release an R package named MAPS on GitHub (https://github.com/caofff/MAPS), providing ready‑to‑use implementations of the static and adaptive algorithms, facilitating adoption by the broader Bayesian computation community.