Making Steppingstones out of Stumbling Blocks: A Bayesian Model Evidence Estimator with Application to Groundwater Transport Model Selection
Making Steppingstones out of Stumbling Blocks: A Bayesian Model Evidence Estimator with Application to Groundwater Transport Model Selection
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DOI：
10.3390/w11081579
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发表时间：
2019-07
期刊：
影响因子：
3.4
通讯作者：
A. Elshall;M. Ye
中科院分区：
文献类型：
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作者：
A. Elshall;M. Ye
贝叶斯模型证据（Bayesian Model Evidence，BME）是一种模型对观测数据的平均拟合的度量，给出了模型可以假设的所有参数值。通过考虑拟合优度和模型复杂性之间的权衡，BME用于模型选择和模型平均目的。对于严格的贝叶斯计算，理论上无偏的蒙特卡罗数值估计优于半解析解。本研究探讨五个BME数值估计，并询问如何准确估计的BME是重要的惩罚模型的复杂性。数值BME估计的极限情况是先验抽样算术平均估计（AM）和后验抽样调和平均（HM）估计，这是直接实现的，但它们分别导致低估和高估。我们还考虑了热力学积分（TI）和阶梯采样（SS）的路径采样方法，该方法对连接先验和后验的多个中间分布进行采样。虽然TI和SS在理论上是无偏估计量，但它们在实际应用中可能会因数值实现而产生偏差。例如，一些中间分布的抽样误差可能会引入偏差。我们提出了一个变种的SS，即多个一步石抽样（MOSS），是不太敏感的抽样误差。我们使用地下水传输模型选择问题来评估这五个估计值。SS和MOSS以有效的计算成本给出了最小偏差的BME估计。如果估计的BME具有与真实BME协变的偏差，这将不是问题，因为我们感兴趣的是BME比率而不是它们的绝对值。相反，结果表明，BME估计偏差可以是模型复杂性的函数。因此，有偏BME估计导致更复杂模型的不准确惩罚，这改变了模型排名。与其他三种方法相比，SS和MOSS观察到的情况较少。
Bayesian model evidence (BME) is a measure of the average fit of a model to observation data given all the parameter values that the model can assume. By accounting for the trade-off between goodness-of-fit and model complexity, BME is used for model selection and model averaging purposes. For strict Bayesian computation, the theoretically unbiased Monte Carlo based numerical estimators are preferred over semi-analytical solutions. This study examines five BME numerical estimators and asks how accurate estimation of the BME is important for penalizing model complexity. The limiting cases for numerical BME estimators are the prior sampling arithmetic mean estimator (AM) and the posterior sampling harmonic mean (HM) estimator, which are straightforward to implement, yet they result in underestimation and overestimation, respectively. We also consider the path sampling methods of thermodynamic integration (TI) and steppingstone sampling (SS) that sample multiple intermediate distributions that link the prior and the posterior. Although TI and SS are theoretically unbiased estimators, they could have a bias in practice arising from numerical implementation. For example, sampling errors of some intermediate distributions can introduce bias. We propose a variant of SS, namely the multiple one-steppingstone sampling (MOSS) that is less sensitive to sampling errors. We evaluate these five estimators using a groundwater transport model selection problem. SS and MOSS give the least biased BME estimation at an efficient computational cost. If the estimated BME has a bias that covariates with the true BME, this would not be a problem because we are interested in BME ratios and not their absolute values. On the contrary, the results show that BME estimation bias can be a function of model complexity. Thus, biased BME estimation results in inaccurate penalization of more complex models, which changes the model ranking. This was less observed with SS and MOSS as with the three other methods.