Although there is much recent work developing flexible variational methods for Bayesian computation, Gaussian approximations with structured covariance matrices are often preferred computationally in high-dimensional settings. This paper considers approximate inference methods for complex latent variable models where the posterior is close to Gaussian, but with some skewness in the posterior marginals. We consider skew decomposable graphical models (SDGMs), which are based on the closed skew normal family of distributions, as variational approximations. These approximations can reflect the true posterior conditional independence structure and capture posterior skewness. Different parametrizations are explored for this variational family, and the speed of convergence and quality of the approximation can depend on the parametrization used. To increase flexibility, implicit copula SDGM approximations are also developed, where elementwise transformations of an approximately standardized SDGM random vector are considered. Our parametrization of the implicit copula approximation is novel, even in the special case of a Gaussian approximation. Performance of the methods is examined in a number of real examples involving generalized linear mixed models and state space models, and we conclude that our copula approaches are most accurate, but that the SDGM methods are often nearly as good and have lower computational demands.

翻译：尽管近期有大量工作开发灵活的变分方法用于贝叶斯计算，但在高维设置中，具有结构化协方差矩阵的高斯近似在计算上往往更受青睐。本文针对复杂潜变量模型提出近似推断方法，其中后验分布接近高斯分布，但后验边缘分布存在一定偏态。我们考虑基于闭偏态正态分布族的偏态可分解图模型（SDGM）作为变分近似。这些近似能够反映真实后验条件独立结构并捕捉后验偏态。我们探索了该变分族的不同参数化方法，收敛速度和近似质量可能取决于所采用的参数化形式。为增加灵活性，还发展了隐式连接函数SDGM近似，其中对近似标准化的SDGM随机向量进行逐元素变换。我们的隐式连接函数近似参数化具有新颖性，即使在高斯近似的特例中也是如此。通过多个涉及广义线性混合模型和状态空间模型的实际案例检验了这些方法的性能，结论表明我们的连接函数方法精度最高，但SDGM方法通常几乎同样准确且计算成本更低。