We present a non-factorized variational method for full posterior inference in Bayesian hierarchical models, with the goal of capturing the posterior variable dependencies via efficient and possibly parallel computation. Our approach unifies the integrated nested Laplace approximation (INLA) under the variational framework. The proposed method is applicable in more challenging scenarios than typically assumed by INLA, such as Bayesian Lasso, which is characterized by the non-differentiability of the $\ell_{1}$ norm arising from independent Laplace priors. We derive an upper bound for the Kullback-Leibler divergence, which yields a fast closed-form solution via decoupled optimization. Our method is a reliable analytic alternative to Markov chain Monte Carlo (MCMC), and it results in a tighter evidence lower bound than that of mean-field variational Bayes (VB) method.
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