Variational approximations provide fast, deterministic alternatives to Markov chain Monte Carlo for Bayesian inference on the parameters of complex, hierarchical models. Variational approximations are often limited in practicality in the absence of conjugate posterior distributions. Recent work has focused on the application of variational methods to models with only partial conjugacy, such as in semiparametric regression with heteroscedastic errors. Here, both the mean and log variance functions are modeled as smooth functions of covariates. For this problem, we derive a mean field variational approximation with an embedded Laplace approximation to account for the nonconjugate structure. Empirical results with simulated and real data show that our approximate method has significant computational advantages over traditional Markov chain Monte Carlo; in this case, a delayed rejection adaptive Metropolis algorithm. The variational approximation is much faster and eliminates the need for tuning parameter selection, achieves good fits for both the mean and log variance functions, and reasonably reflects the posterior uncertainty. We apply the methods to log-intensity data from a small angle X-ray scattering experiment, in which properly accounting for the smooth heteroscedasticity leads to significant improvements in posterior inference for key physical characteristics of an organic molecule.

中文翻译：

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存在异方差误差时半参数回归的拉普拉斯变分逼近

变分近似提供了马尔可夫链蒙特卡罗的快速、确定性替代方案，用于对复杂、分层模型的参数进行贝叶斯推理。在没有共轭后验分布的情况下，变分近似在实用性上通常是有限的。最近的工作重点是将变分方法应用于仅具有部分共轭的模型，例如在具有异方差误差的半参数回归中。在这里，均值和对数方差函数都被建模为协变量的平滑函数。对于这个问题，我们使用嵌入的拉普拉斯近似推导出平均场变分近似来解释非共轭结构。模拟和真实数据的实证结果表明，我们的近似方法比传统的马尔可夫链蒙特卡罗方法具有显着的计算优势；在这种情况下，延迟拒绝自适应 Metropolis 算法。变分逼近速度更快，无需调整参数选择，对均值和对数方差函数都实现了良好的拟合，并合理地反映了后验不确定性。我们将这些方法应用于来自小角度 X 射线散射实验的对数强度数据，在该实验中，正确考虑平滑异方差性会显着改善有机分子关键物理特性的后验推断。变分逼近速度更快，无需调整参数选择，对均值和对数方差函数都实现了良好的拟合，并合理地反映了后验不确定性。我们将这些方法应用于来自小角度 X 射线散射实验的对数强度数据，在该实验中，正确考虑平滑异方差性会显着改善有机分子关键物理特性的后验推断。变分逼近速度更快，无需调整参数选择，对均值和对数方差函数都实现了良好的拟合，并合理地反映了后验不确定性。我们将这些方法应用于来自小角度 X 射线散射实验的对数强度数据，在该实验中，适当考虑平滑异方差性会显着改善对有机分子关键物理特性的后验推断。