🤖 AI Summary
This work addresses the inaccuracy of posterior approximations in variational inference caused by limited expressiveness of conventional variational families. To overcome this limitation, the authors propose a novel variational method that integrates wavelet basis representations with copula-based dependency structures. The approach leverages discrete wavelet transforms to flexibly capture complex marginal posterior densities and explicitly models high-dimensional dependencies among parameters using copula functions. Efficient inference is achieved through Monte Carlo estimation of the evidence lower bound (ELBO), automatic differentiation, and gradient-based optimization. As the first method to jointly employ wavelets and copulas in variational inference, it demonstrates competitive posterior mean estimates—on par with Markov chain Monte Carlo (MCMC)—and substantially improved uncertainty quantification over standard variational approaches across logistic regression, sparse linear models, and hierarchical models.
📝 Abstract
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.