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.

翻译：在由微分方程控制动态的纵向模型中，参数估计可能需要与数值积分同时进行。我们发现了一种情况，其中数值积分引入误差的方式成为估计中非唯一性的新来源。我们得到了两组差异极大的参数，一组是对真实值的良好估计，另一组则非常不准确。由于数值误差，这两组估计的前向数值投影在统计上无法区分。在这种情况下，参数的后验分布呈现双峰形态：一个主导模态更接近真实参数值，另一个簇则围绕错误值聚集。我们通过仿射一阶微分方程从理论和实证两方面证明了多模态的存在性，提出仿真工作流可在更一般情形下检测该问题的证据，并表明采用适当解决方案的马尔可夫链蒙特卡洛采样能够避免双峰性。这种由数值误差导致的多模态后验问题，对更广泛依赖数值积分的贝叶斯反演方法具有重要影响。