CONGA: Compiling Gaussian Fiber Geometry for Structured Bayesian Inference

Many structured Bayesian models contain posterior geometry that generic inference methods must relearn during warmup. We consider a high-dimensional block that is conditionally Gaussian given a smaller non-Gaussian core. Its conditional mean and precision define a family of exact affine transports, including symmetric Brenier and graph-sparse Knothe–Rosenblatt maps. These transports factorize the target into a standard normal block and the collapsed core, so KL and relative Fisher training objectives depend only on the core. We introduce CONGA, a compiler that finds candidate Gaussian blocks, checks their precision structure, and chooses a temporal, spectral, or sparse implementation for the given workload. A numerical check alone does not justify marginalization: without an exact model-level identity, CONGA either applies a target-preserving change of variables or leaves the model unchanged. On the Stock–Watson inflation model, it improves minimum ESS per second by $12.5\times$ for NUTS and $23.7\times$ for WALNUTS. On a 6,760-dimensional spatio-temporal load model, it reduces sampling to a 34-dimensional core and improves the same metric by $234\times$. Experiments with graph-indexed fields show that the best exact map also depends on coloring and fill, and that the original target can sometimes remain the better choice.