Solving the Schrödinger Bridge via Bayesian Inference in Parameter Space

Solving transport problems, i.e., finding a map that transports one given distribution to another, has numerous applications in machine learning. Currently, the Schrödinger Bridge (SB) is commonly used to describe the mapping between two arbitrary distributions. However, most existing SB solutions either suffer from slow convergence and sampling due to the difficulty of directly optimizing the likelihood or are highly sensitive to initialization and hyperparameters. To address these, we propose a novel Bayesian Schrödinger Bridge (BSB) formalism to address the SB problem by explicit likelihood factorization leveraging Bayesian inference. By explicitly introducing Bayesian inference in the continuous parameter space of a Gaussian mixture model, BSB allows tractable and explicit likelihood-based optimization of the likelihood instead of the variational lower bound, while enabling more accurate modeling of complex probability density transitions. BSB analytically derives closed-form Bayesian inference at each time step, enabling the design of SDE and higher-order ODE solvers for few-step high-quality generation. Extensive experiments validate our effectiveness, achieving faster convergence, robust training, and high-quality few-step SB solution.