Stat Colloquium: Dr. Debangan Dey

Title: Bigraphical Matérn–Whittle processes for fast inference of big multivariate spatial data on general domains

Abstract: Modern spatial data sets record many variables at many locations, on domains with boundaries, holes and compartments. A spatial transcriptomics experiment, for example, measures over a thousand genes in tens of thousands of cells across a tissue section. Joint modelling of all variables is essential for prediction and for learning which variables depend on which, yet the exact Gaussian process likelihood costs cubic time in the number of variable and location pairs, classical cross-covariance constructions couple every pair of variables, and Euclidean distance is the wrong notion of proximity on such domains. We introduce the bigraphical Matérn–Whittle (BMW) process, a multivariate Gaussian process built from two graphs. A nearest-neighbour graph on the locations supplies a spatial operator through the Whittle stochastic partial differential equation, and a directed acyclic graph on the variables couples the fields through a block lower-triangular operator system. The construction is valid for any domain a graph can describe and for any coefficients, with no stability constraint. Absent edges correspond to exact process-level conditional independence, so the learned graph is a scientific object rather than a computational device. The triangular structure makes the prior log-determinant a closed-form spectral sum that is free of the cross-dependence coefficients, and together with Lanczos iterations, stochastic quadrature and conjugate gradient solves, every step of estimation, prediction and Bayesian graph selection is linear in the data size with the model never approximated. A fast variational stage is followed by an exact likelihood refinement that removes the mean-field bias. On a MERFISH mouse brain section with twenty-two million pairs, the model is fitted in seventy-five minutes on a laptop, reduces held-out prediction error for myelin genes by up to ninety-one percent, and recovers the oligodendrocyte gene module without supervision. An open-source Python package, bmwspatial, implements the method.