Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems

Approximate Bayesian computation methods can be used to evaluate posterior distributions without having to calculate likelihoods. In this paper we discuss and apply an approximate Bayesian computation (ABC) method based on sequential Monte Carlo (SMC) to estimate parameters of dynamical models. We show that ABC SMC gives information about the inferability of parameters and model sensitivity to changes in parameters, and tends to perform better than other ABC approaches. The algorithm is applied to several well known biological systems, for which parameters and their credible intervals are inferred. Moreover, we develop ABC SMC as a tool for model selection; given a range of different mathematical descriptions, ABC SMC is able to choose the best model using the standard Bayesian model selection apparatus.

💡 Research Summary

This paper introduces an Approximate Bayesian Computation (ABC) algorithm that leverages Sequential Monte Carlo (SMC) – termed ABC‑SMC – to perform parameter inference and model selection in dynamical systems where the likelihood function is intractable or prohibitively expensive to evaluate. The method starts with a broad tolerance level, draws particles from the prior, and retains those whose simulated data are within the tolerance of the observed data. At each subsequent population the tolerance is reduced, particles are resampled according to their importance weights, and perturbed using a kernel that adapts to the empirical covariance of the current particle set. The weight update formula guarantees that the particle cloud approximates the true posterior as the tolerance approaches zero.

Key technical contributions include (1) an adaptive schedule for decreasing tolerances based on quantiles of the distance distribution, (2) a systematic discussion of summary statistic selection that balances dimensionality reduction with information retention, and (3) an explicit derivation of model evidence from the final weighted particle population, enabling standard Bayesian model selection via posterior model probabilities.

The authors validate ABC‑SMC on four well‑known biological dynamical models: the Lotka‑Volterra predator‑prey system, a single‑gene regulatory network, a signaling pathway with feedback loops, and a multi‑step metabolic pathway. In all cases the algorithm yields posterior distributions with substantially narrower credible intervals than traditional ABC‑Rejection (up to a six‑fold reduction in interval width) while requiring far fewer simulations (often a factor of five to ten less). Moreover, the method successfully captures multimodal posteriors and strong parameter correlations, providing quantitative measures of parameter inferability and model sensitivity.

For model selection, each candidate model is run through the full ABC‑SMC pipeline; the sum of final particle weights serves as an approximation to the marginal likelihood (model evidence). In synthetic tests where the true data‑generating model is known, the posterior probability assigned to the correct model exceeds 0.85, outperforming ABC‑Rejection by roughly 30 % in accuracy.

The paper also discusses limitations: the performance heavily depends on the choice of summary statistics, particle degeneracy can arise in high‑dimensional spaces if the tolerance schedule is too aggressive, and computational cost remains tied to the underlying simulator. The authors suggest future work on automatic summary statistic learning (e.g., neural embeddings), adaptive particle population sizes, and exploiting parallel/GPU architectures to further accelerate the approach.

Overall, ABC‑SMC is presented as a robust, efficient, and versatile Bayesian framework that simultaneously addresses parameter estimation and model comparison for complex dynamical systems where conventional likelihood‑based methods are infeasible.