We propose a framework for computing, optimizing and integrating with respect to a smooth marginal likelihood in statistical models that involve high-dimensional parameters/latent variables and continuous low-dimensional hyperparameters. The method requires samples from the posterior distribution of the parameters for different values of the hyperparameters on a simulation grid and returns inference on the marginal likelihood defined everywhere on its domain, and on its functionals. We show how the method relates to many of the methods that have been used in this context, including sequential Monte Carlo, Gibbs sampling, Monte Carlo maximum likelihood, and umbrella sampling. We establish the consistency of the proposed estimators as the sampling effort increases, both when the simulation grid is kept fixed and when it becomes dense in the domain. We showcase the approach on Gaussian process regression and classification and crossed effect models.

翻译：我们提出了一种框架，用于在涉及高维参数/潜变量和连续低维超参数的统计模型中计算、优化和积分平滑的边际似然。该方法需要在模拟网格上针对不同超参数值获取参数后验分布的样本，并返回在其定义域上任意点定义的边际似然及其泛函的推断。我们展示了该方法如何关联到在此背景下使用的多种方法，包括序贯蒙特卡洛、吉布斯采样、蒙特卡洛最大似然以及伞形采样。我们证明了当采样量增加时，所提出估计量的一致性，包括在模拟网格保持固定和在其定义域内变得稠密两种情况。我们在高斯过程回归与分类以及交叉效应模型上展示了该方法的应用。