Variational inference (VI) has become a widely used approach for scalable Bayesian inference, but its performance strongly depends on the flexibility of the chosen variational family. In this work, we propose a novel variational family that combines wavelet-based representations for marginal posterior densities with copula functions to model dependence structures. The marginal distributions are constructed using coefficients from the discrete wavelet transform, providing a flexible and adaptive framework capable of capturing complex features such as asymmetry. The joint distribution is then obtained through a copula, allowing for explicit modeling of dependence among parameters, including both independence and Gaussian copula structures. We develop an efficient estimation procedure based on Monte Carlo approximations of the evidence lower bound (ELBO) and automatic differentiation, enabling scalable optimization using gradient-based methods. Through extensive simulation studies, including logistic regression, sparse linear models, and hierarchical models, we demonstrate that the proposed approach achieves posterior mean estimates comparable to Markov chain Monte Carlo (MCMC) methods, while providing improved uncertainty quantification relative to standard variational approaches. Applications to hierarchical logistic regression and Bayesian conditional transformation models further illustrate the practical advantages of the method in complex, high dimensional settings. The proposed wavelet copula variational family offers a flexible and computationally efficient alternative for Bayesian inference.

翻译：变分推断（VI）已成为可扩展贝叶斯推断的常用方法，但其性能很大程度上依赖于所选变分族的灵活性。本文提出一种新型变分族，将基于小波变换的边缘后验密度表示与Copula函数相结合以建模依赖结构。通过离散小波变换系数构建边缘分布，提供灵活且自适应的框架，能够捕捉非对称性等复杂特征。进一步利用Copula获得联合分布，从而显式建模参数间的依赖关系，包括独立和高斯Copula结构。我们开发了基于蒙特卡洛近似的证据下界（ELBO）与自动微分的高效估计流程，实现了基于梯度方法的可扩展优化。通过大量模拟研究（包括逻辑回归、稀疏线性模型和层次模型），证明所提方法得到的后验均值估计与马尔可夫链蒙特卡洛（MCMC）方法相当，同时相比标准变分方法提供了更优的不确定性量化。在层次逻辑回归与贝叶斯条件变换模型中的应用进一步展示了该方法在复杂高维场景下的实际优势。所提出的小波Copula变分族为贝叶斯推断提供了一种灵活且计算高效的替代方案。