AI 中文总结
针对高维潜在结构贝叶斯模型计算难题，提出KNorm-RVB框架，通过归一化和偏度降低重参数化局部变量条件后验，结合高斯与偏态正态变分族，在多种模型中提升后验近似精度。
AI 中文摘要
具有高维潜在结构的贝叶斯层次模型需要可扩展的后验近似，既要保留关键依赖关系，又要保持计算上的易处理性。平均场变分推理（MFVI）效率高，但当局部变量与全局变量高度相关或紧密耦合时可能不可靠。我们提出了KNorm-RVB，这是一种用于具有稀疏局部精度矩阵的潜在高斯和潜在非高斯模型的广义重参数化变分贝叶斯框架。KNorm-RVB通过归一化和随后的偏度降低，将局部变量的条件后验映射到标准高斯分布，这由一种新颖的K分量反对称密度表示实现。这种重参数化将变换后的条件局部后验集中在一个优化的反射点，并使局部和全局变量去相关，使MFVI更有效。在对称条件下，我们表明MFVI能精确恢复局部后验均值和相关矩阵，这促使在应用MFVI之前对条件局部后验进行KNorm-RVB的归一化和对称化。我们将用于重参数化局部变量的高斯变分族与用于其余变量的灵活封闭偏态正态族相结合。在广义线性混合模型、混合多项逻辑模型、空间自回归模型和随机波动率模型中，KNorm-RVB比现有方法提高了后验近似精度。
英文摘要
Bayesian hierarchical models with high-dimensional latent structure require scalable posterior approximations that preserve key dependencies while remaining computationally tractable. Mean-field variational inference (MFVI) is efficient, but can be unreliable when local variables are strongly correlated or tightly coupled to global variables. We propose KNorm-RVB, a generalized reparametrized variational Bayes framework for latent Gaussian and latent non-Gaussian models with sparse local precision matrices. KNorm-RVB maps the conditional posterior of local variables toward a standard Gaussian via normalization followed by skewness reduction, enabled by a novel K-component skew-symmetric density representation. This reparametrization centers the transformed conditional local posterior at an optimized reflection point and decorrelates local and global variables, making MFVI much more effective. Under symmetry conditions, we show that MFVI recovers the local posterior mean and correlation matrix exactly, motivating KNorm-RVB's normalization and symmetrization of the conditional local posterior before applying MFVI. We combine a Gaussian variational family for reparametrized local variables with a flexible closed skew normal family for the remaining variables. Across generalized linear mixed models, mixed multinomial logit models, spatial autoregressive models, and stochastic volatility models, KNorm-RVB improves posterior approximation accuracy over existing methods.
Comments48 pages, 9 figures
