  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.
