In order to quantitatively predict nano- as well as other particle-size distributions, one needs to have both a mathematical model and estimates of the parameters that appear in these models. Here, we show how one can use Bayesian inversion to obtain statistical estimates for the parameters that appear in recently derived mechanism-enabled population balance models (ME-PBM) of nanoparticle growth. The Bayesian approach addresses the question of “how well do we know our parameters, along with their uncertainties?.” The results reveal that Bayesian inversion statistical analysis on an example, prototype nanoparticle formation system allows one to estimate not just the most likely rate constants and other parameter values, but also their SDs, confidence intervals, and other statistical information. Moreover, knowing the reliability of the mechanistic model's parameters in turn helps inform one about the reliability of the proposed mechanism, as well as the reliability of its predictions. The paper can also be seen as a tutorial with the additional goal of achieving a “Gold Standard” Bayesian inversion ME-PBM benchmark that others can use as a control to check their own use of this methodology for other systems of interest throughout nature. Overall, the results provide strong support for the hypothesis that there is substantial value in using a Bayesian inversion methodology for parameter estimation in particle formation systems.

中文翻译：

---

在纳米粒子尺寸分布的机制启用的人口平衡模型中估计反应参数：贝叶斯逆问题方法

为了定量预测纳米以及其他粒度分布，需要同时拥有数学模型和这些模型中出现的参数的估计值。在这里，我们展示了如何使用贝叶斯反演来获得参数的统计估计值，这些参数出现在最近衍生的纳米粒子生长的机制启用人口平衡模型 (ME-PBM) 中。贝叶斯方法解决了“我们对参数及其不确定性的了解程度如何？”的问题。结果表明，贝叶斯反演统计分析示例，原型纳米粒子形成系统不仅可以估计最可能的速率常数和其他参数值，还可以估计它们的 SD、置信区间和其他统计信息。此外，了解机械模型参数的可靠性反过来有助于告知人们所提出机制的可靠性以及其预测的可靠性。该论文也可以被视为一个教程，其额外目标是实现“黄金标准”贝叶斯反演 ME-PBM 基准，其他人可以使用该基准作为对照来检查他们自己对该方法的使用情况，以检查整个自然界中其他感兴趣的系统。总体而言，结果为使用贝叶斯反演方法在粒子形成系统中进行参数估计具有重要价值的假设提供了强有力的支持。