In computational mechanics, multiple models are often present to describe a physical system. While Bayesian model selection is a helpful tool to compare these models using measurement data, it requires the computationally expensive estimation of a multidimensional integral -- known as the marginal likelihood or as the model evidence (\textit{i.e.}, the probability of observing the measured data given the model). This study presents efficient approaches for estimating this marginal likelihood by transforming it into a one-dimensional integral that is subsequently evaluated using a quadrature rule at multiple adaptively-chosen iso-likelihood contour levels. Three different algorithms are proposed to estimate the probability mass at each adapted likelihood level using samples from importance sampling, stratified sampling, and Markov chain Monte Carlo sampling, respectively. The proposed approach is illustrated through four numerical examples. The first example validates the algorithms against a known exact marginal likelihood. The second example uses an 11-story building subjected to an earthquake excitation with an uncertain hysteretic base isolation layer with two models to describe the isolation layer behavior. The third example considers flow past a cylinder when the inlet velocity is uncertain. Based on these examples, the method with stratified sampling is by far the most accurate and efficient method for complex model behavior in low dimension. In the fourth example, the proposed approach is applied to heat conduction in an inhomogeneous plate with uncertain thermal conductivity modeled through a 100 degree-of-freedom Karhunen-Lo\`{e}ve expansion. The results indicate that MultiNest cannot efficiently handle the high-dimensional parameter space, whereas the proposed MCMC-based method more accurately and efficiently explores the parameter space.

翻译：在计算力学中，通常存在多个模型来描述一个物理系统。虽然贝叶斯模型选择是一种利用测量数据比较这些模型的有用工具，但它需要计算成本高昂的多维积分估计——即边缘似然或模型证据（即给定模型下观测到测量数据的概率）。本研究提出了通过将边缘似然转化为一维积分来高效估计它的方法，该一维积分随后在多个自适应选择的等似然轮廓水平上使用求积法则进行评估。提出了三种不同的算法，分别利用重要性采样、分层采样和马尔可夫链蒙特卡洛采样的样本来估计每个自适应似然水平的概率质量。通过四个数值算例对所提方法进行了说明。第一个算例针对已知精确边缘似然验证了算法。第二个算例使用一个受到地震激励的11层建筑，其具有不确定的滞回基础隔震层，并采用两个模型来描述隔震层行为。第三个算例考虑入口速度不确定时圆柱绕流问题。基于这些算例，对于低维复杂模型行为，采用分层采样的方法是目前最准确且高效的方法。在第四个算例中，所提方法被应用于具有不确定热导率的不均匀平板热传导问题，该热导率通过一个100自由度的Karhunen-Loève展开进行建模。结果表明，MultiNest无法高效处理高维参数空间，而所提出的基于MCMC的方法能够更准确、更高效地探索参数空间。