Besag-York-Mollie Model

The Besag-York-Mollie (BYM) model is the workhorse hierarchical Bayesian model for small-area disease mapping. Proposed by Julian Besag, Jeremy York, and Annie Mollie (1991), it models area-level disease counts with a Poisson likelihood whose log relative risk is the sum of two random effects: a spatially structured component, given an intrinsic conditional autoregressive (ICAR) prior that borrows strength from neighboring areas, and an unstructured component capturing area-specific heterogeneity that is not spatially patterned. This convolution of structured and unstructured effects lets the model smooth noisy small-area rates toward local and global means while distinguishing genuine spatial trend from independent overdispersion. Because the original parameterization makes the two variance components hard to interpret and depends on the graph, Riebler, Sorbye, Simpson, and Rue (2016) introduced the scaled BYM2 reparameterization, which mixes a scaled spatial effect and an unstructured effect through a single interpretable mixing parameter and a total-variance parameter, improving prior specification and identifiability.