论文标题
增强的拉普拉斯近似
Enhanced Laplace Approximation
论文作者
论文摘要
已提出了拉普拉斯近似（LA）作为近似具有潜在变量的统计模型的边际可能性的方法。但是，基于LA的近似最大似然估计器（MLE）通常会偏向二进制或空间数据，相应的Hessian矩阵低估了这些近似MLE的标准误差。已经提出了高阶近似值；但是，它不能应用于复杂的模型，例如相关的随机效应模型，也不能提供一致的方差估计器。在本文中，我们提出了一个增强的LA（ELA），该LA（ELA）提供了真正的MLE及其一致方差估计器。我们研究了它与变分贝叶斯方法的关系。我们还引入了一个新的受限制的最大似然估计器（REMLE），以估计分散参数。数值研究的结果表明，ELA提供了令人满意的MLE和REMLE，以及它们的固定参数方差估计器。 MLE和REMLE分别可以看作是平坦先验下的后验模式和边缘后部模式。一些比较也与不同先验的贝叶斯程序进行。
The Laplace approximation (LA) has been proposed as a method for approximating the marginal likelihood of statistical models with latent variables. However, the approximate maximum likelihood estimators (MLEs) based on the LA are often biased for binary or spatial data, and the corresponding Hessian matrix underestimates the standard errors of these approximate MLEs. A higher-order approximation has been proposed; however, it cannot be applied to complicated models such as correlated random effects models and does not provide consistent variance estimators. In this paper, we propose an enhanced LA (ELA) that provides the true MLE and its consistent variance estimator. We study its relationship to the variational Bayes method. We also introduce a new restricted maximum likelihood estimator (REMLE) for estimating dispersion parameters. The results of numerical studies show that the ELA provides a satisfactory MLE and REMLE, as well as their variance estimators for fixed parameters. The MLE and REMLE can be viewed as posterior mode and marginal posterior mode under flat priors, respectively. Some comparisons are also made with Bayesian procedures under different priors.
