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	Besag-York-Mollie Model	Poisson Rate Regression
Field	Spatial Epidemiology	Social Epidemiology
Family	Regression model	Regression model
Year of origin	1991	1983
Originator	Julian Besag, Jeremy York & Annie Mollie (BYM2 by Riebler, Sorbye, Simpson & Rue)	E. L. Frome (rate formulation); A. C. Cameron & P. K. Trivedi (modern count-data treatment)
Type	Hierarchical Bayesian Poisson model with structured and unstructured spatial random effects	Generalized linear model for event rates and counts with log link and person-time offset
Seminal source	Besag, J., York, J., & Mollie, A. (1991). Bayesian image restoration, with two applications in spatial statistics. Annals of the Institute of Statistical Mathematics, 43(1), 1-20.	Frome, E. L. (1983). The Analysis of Rates Using Poisson Regression Models. Biometrics, 39(3), 665-674.
Aliases	BYM Model, Convolution Prior Model, CAR Convolution Model, BYM2 Reparameterization	Poisson Regression for Rates, Log-Linear Rate Model, Incidence-Rate-Ratio Regression, Poisson Regression with Offset
Related	4	3
Summary	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.	Poisson rate regression is the standard generalized linear model for analyzing event rates and counts, such as the number of deaths, hospitalizations, or new cases observed over a span of person-time. It models the logarithm of the expected event rate as a linear function of covariates, using a Poisson likelihood and a log link, and accommodates differing amounts of exposure by including the log of person-time as an offset. Because coefficients enter on the log scale, their exponentials are incidence-rate ratios that quantify multiplicative effects on the rate. The rate formulation was crystallized in Frome's 1983 Biometrics paper, and the model sits within the broader count-data framework developed comprehensively by Cameron and Trivedi, who also detail its central practical concern: overdispersion, where the variance exceeds the Poisson assumption and standard errors must be corrected.