ADVANCES IN BAYESIAN METHODS FOR QUANTITATIVE GENETIC ANALYSIS Daniel Gianola 1 , Romdhane Rekaya 2 , Guilherme J. M. Rosa 3 and Adhemar Sanches 4 1 Department of Animal Sciences, University of Wisconsin-Madison, WI 53706, USA 2 Department of Animal and Dairy Science, University of Georgia, Athens, GA 30602, USA 3 Departments of Animal Sciences and of Fisheries & Wildlife, Michigan State University, East Lansing, MI 48824-1225, USA 4 Departamento de Ciencias Exatas, FCAV/UNESP, 14870-000 Jaboticabal SP, Brasil INTRODUCTION Arguably, the Bayesian controversy, in the sense of Blasco (2001), is over or will be over soon. Bayesian methods are employed now routinely in archaeology, artificial intelligence, disease mapping, ecology, economics, forest management, physics, signal processing and last, but not least, quantitative genetics and genomics. For example, Shoemaker et al. (1999) write: Bayesian approaches can be easier to interpret and they have been employed in many genetic areas, including: the classification of genotypes and estimating relationships, population genetics and molecular evolution, linkage mapping (including gene ordering and quantitative risk analysis) and quantitative genetics [including quantitative trait (QTL) mapping]. This should be augmented to include transcriptional analysis (Newton et al., 2001). Walsh (2001), in a discussion of quantitative genetics in the age of genomics, has conjectured that, in the next 20 years, Bayesian procedures will likely displace their likelihood based-counterparts. This is gratifying to some animal breeders that have been advocating the Bayesian approach for 20 years or so (e.g., Gianola and Foulley, 1982; Gianola and Fernando, 1986). Further, and particularly in connection with