Here Be Dragons: Bimodal posteriors arise from numerical integration error in longitudinal models

Longitudinal models with dynamics governed by differential equations may require numerical integration alongside parameter estimation. We have identified a situation where the numerical integration introduces error in such a way that it becomes a novel source of non-uniqueness in estimation. We obtain two very different sets of parameters, one of which is a good estimate of the true values and the other a very poor one. The two estimates have forward numerical projections statistically indistinguishable from each other because of numerical error. In such cases, the posterior distribution for parameters is bimodal, with a dominant mode closer to the true parameter value, and a second cluster around the errant value. We demonstrate that multi-modality exists both theoretically and empirically for an affine first order differential equation, that a simulation workflow can test for evidence of the issue more generally, and that Markov Chain Monte Carlo sampling with a suitable solution can avoid bimodality. The issue of multi-modal posteriors arising from numerical error has consequences for Bayesian inverse methods that rely on numerical integration more broadly.

💡 Research Summary

The paper “Here Be Dragons: Bimodal posteriors arise from numerical integration error in longitudinal models” investigates a previously under‑explored source of non‑identifiability in Bayesian inference for longitudinal differential‑equation models: the numerical integration routine itself. The authors focus on a simple linear first‑order ordinary differential equation (ODE)

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