In the Bayesian framework, the marginal likelihood plays an important role in variable selection and model comparison. The marginal likelihood is the marginal density of the data after integrating out the parameters over the parameter space. However, this quantity is often analytically intractable due to the complexity of the model. In this paper, we first examine the properties of the inflated density ratio (IDR) method, which is a Monte Carlo method for computing the marginal likelihood using a single MC or Markov chain Monte Carlo (MCMC) sample. We then develop a variation of the IDR estimator, called the dimension reduced inflated density ratio (Dr.IDR) estimator. We further propose a more general identity and then obtain a general dimension reduced (GDr) estimator. Simulation studies are conducted to examine empirical performance of the IDR estimator as well as the Dr.IDR and GDr estimators. We further demonstrate the usefulness of the GDr estimator for computing the normalizing constants in a case study on the inequality-constrained analysis of variance.

中文翻译：

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膨胀密度比及其变化和泛化，用于计算边缘似然

在贝叶斯框架中，边际可能性在变量选择和模型比较中起着重要作用。边际可能性是在参数空间上对参数进行积分之后数据的边际密度。但是，由于模型的复杂性，此数量通常在分析上难以处理。在本文中，我们首先检查了膨胀密度比（IDR）方法的属性，该方法是使用单个MC或Markov链蒙特卡洛（MCMC）样本计算边际似然性的蒙特卡洛方法。然后，我们开发IDR估算器的一种变体，称为降维膨胀密度比（Dr.IDR）估算器。我们进一步提出了一个更一般的恒等式，然后获得了广义降维（GDr）估计量。进行模拟研究以检查IDR估计器以及Dr.IDR和GDr估计器的经验性能。我们进一步证明了在不等式约束方差分析的案例研究中，GDr估计器对于计算归一化常数的有用性。