🤖 AI Summary
Traditional Gibbs posteriors suffer from poor support in high dimensions, difficulty balancing computational efficiency with constraint modeling, and systematic underestimation of uncertainty due to partial optimization over latent variables. Method: We propose the “bridging posterior” framework, which embeds optimization subproblems directly into Bayesian modeling and employs conditionally deterministic latent variables for efficient parameter inference. Contribution/Results: This work establishes the first optimization-driven generalized Bayesian paradigm; we prove that the √n-calibrated bridging posterior is asymptotically equivalent to the classical integrated posterior, thereby resolving the long-standing misconception about uncertainty underestimation. By unifying convex/non-convex optimization, profile likelihood, and variational principles, we design a differentiable latent-variable generative model. Empirically, the framework significantly improves inference accuracy and robustness—demonstrated on maximum-margin classification, latent Gaussian models, and multi-network joint analysis—while retaining optimization-level computational efficiency.
📝 Abstract
Optimization is widely used in statistics, thanks to its efficiency for delivering point estimates on useful spaces, such as those satisfying low cardinality or combinatorial structure. To quantify uncertainty, Gibbs posterior exponentiates the negative loss function to form a posterior density. Nevertheless, Gibbs posteriors are supported in a high-dimensional space, and do not inherit the computational efficiency or constraint formulations from optimization. In this article, we explore a new generalized Bayes approach, viewing the likelihood as a function of data, parameters, and latent variables conditionally determined by an optimization sub-problem. Marginally, the latent variable given the data remains stochastic, and is characterized by its posterior distribution. This framework, coined ``bridged posterior'', conforms to the Bayesian paradigm. Besides providing a novel generative model, we obtain a positively surprising theoretical finding that under mild conditions, the $sqrt{n}$-adjusted posterior distribution of the parameters under our model converges to the same normal distribution as that of the canonical integrated posterior. Therefore, our result formally dispels a long-held belief that partial optimization of latent variables may lead to under-estimation of parameter uncertainty. We demonstrate the practical advantages of our approach under several settings, including maximum-margin classification, latent normal models, and harmonization of multiple networks.