Online Mean Field Inference of Conditional Random Field

Clustering on large-scale correlated datasets that contain spatial information presents significant statistical and computational challenges. In many domains, such as ultra–high-resolution medical imaging and spatial omics, jointly identifying clusters and coherent spatial patterns is essential for discovering the underlying biological mechanisms. Conditional Random Fields (CRFs) provide a principled probabilistic framework for modeling spatial dependencies. However, conventional CRF inference methods, including Markov chain Monte Carlo and variational inference, are computationally intensive and often infeasible for large or streaming datasets. To address the limitations, we develop an online mean-field variational inference algorithm for CRFs.
Our framework incorporates spatial neighborhood structures into a Bayesian hierarchical model and updates the variational distribution using an efficient online optimization algorithm. The proposed algorithm minimizes the evidence lower bound using online gradient descent, which eliminates the need to load the entire dataset into memory. On large-scale spatial omics datasets, our method accurately captures biologically meaningful spatial patterns while scaling efficiently to millions of spatial locations.