Y. Bard, . Rjc, R. L. Fernando, and D. Gianola, Bayesian inference about dispersion parameters of univariate mixed models with maternal effects: theoretical considerations, Genet Sel Evol, vol.24, pp.107-135, 1974.

G. Casella and E. I. George, Explaining the Gibbs sampler, Am Stat, vol.46, pp.167-174, 1992.

J. J. Colleau, C. Beaumont, and D. Regaldo, Restricted maximum likelihood (REML) estimation of genetic parameters for type traits in Normande cattle breed, Livest Prod Sci, vol.23, pp.47-66, 1989.
URL : https://hal.archives-ouvertes.fr/hal-02720005

J. K. Cullum and R. A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Tlol I: Theory, 1985.

V. Ducrocq and P. Humblot, Genetic characteristics and evolution of semen production of young Normande bulls, Livest Prod Sci, vol.41, pp.1-10, 1995.
URL : https://hal.archives-ouvertes.fr/hal-02701530

V. Ducrocq and G. Casella, A Bayesian analysis of mixed survival models, Genet Sel Evol, vol.28, pp.505-529, 1996.
URL : https://hal.archives-ouvertes.fr/hal-00894151

V. Ducrocq and H. Chapuis, Generalizing the use of the canonical transformation for the solution of multivariate mixed model equations, Genet Sel Evol, vol.29, pp.205-224, 1997.
URL : https://hal.archives-ouvertes.fr/hal-00894164

L. A. Garcia-cortes and D. Sorensen, On a multivariate implementation of the Gibbs sampler, Genet Sel Evol, vol.28, pp.121-126, 1996.
URL : https://hal.archives-ouvertes.fr/hal-00894125

A. E. Gelfand and A. Smith, Sampling-based approaches to calculating marginal densities, J Am Stat Assoc, vol.85, pp.398-409, 1990.

A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis, 1995.

C. J. Geyer, Practical Markov chain Monte-Carlo (with discussion), Stat Sci, vol.7, pp.467-511, 1992.

D. Gianola and R. L. Fernando, Bayesian methods in animal breeding theory, J Anim Sci, vol.63, pp.217-344, 1986.

D. Gianola and J. L. Foulley, Variance estimation from integrated likelihoods (VEIL), 1990.
URL : https://hal.archives-ouvertes.fr/hal-00893855

, Genet Sel Evol, vol.22, pp.403-417

D. Gianola, S. Im, R. L. Fernando, and J. L. Foulley, Mixed model methodology and the Box-Cox theory of transformations: a Bayesian approach, Advances in Statistical Methods for the Genetic Improvement of Livestock, pp.15-40, 1990.
URL : https://hal.archives-ouvertes.fr/hal-02727852

D. Gianola, S. Im, and F. W. Macedo, A framework for prediction of breeding values, Advances in Statistical Methods for the Genetic Improvement of Livestock, pp.211-238, 1990.

W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Introducing Markov chain Monte-Carlo. In: Markov Chain Monte-Carlo in Practice, pp.1-19, 1996.

D. A. Harville, Maximum likelihood approaches to variance component estimation and related problems, J Am Stat Assoc, vol.72, pp.320-338, 1977.

C. R. Henderson, Sire evaluation and genetic trend, Proc Animal Breeding and Genetics Symp in Honor of Dr JL Lush, pp.10-41, 1973.

L. Janss, R. Thompson, and J. Van-arendonk, Application of Gibbs sampling for inference in a mixed major gene -polygenic inheritance model in animal populations, Theor Appl Genet, vol.91, pp.1137-1147, 1995.

J. Makulska, C. Hagger, N. Kfnzi, and H. U. Kupferschmid, Genetic and environmental influences on semen traits in AI bulls, Reprod Dom Anim, vol.28, pp.279-284, 1993.

G. Marsaglia and A. Zaman, Some portable very-long-period random number generators, Comput Phys, vol.8, pp.117-121, 1994.

K. Meyer, Maximum likelihood estimation of variance components for a multivariate mixed model with equal design matrices, Biometrics, vol.41, pp.153-165, 1985.

K. Meyer and S. P. Smith, Restricted maximum likelihood estimation for animal models using derivatives of the likelihood, Genet Sel Evol, vol.28, pp.23-49, 1996.
URL : https://hal.archives-ouvertes.fr/hal-00894119

F. Ouali, Composante g6n6tique de la fonction sexuelle: h6ritabilit6 des caract6res du spermogramme et de la morphologie testiculaire chez les jeunes taurillons de race Montb6liarde, M6moire MSc Vet, 1984.

H. D. Patterson and R. Thompson, Recovery of inter-block information when block sizes are unequal, Biometrika, vol.58, pp.545-554, 1971.

C. Robert and V. Ducrocq, Computation of all eigenvalues of matrices used in restricted maximum likelihood estimation of variance components using sparse matrix techniques, Genet Sel Evol, vol.28, pp.51-65, 1996.
URL : https://hal.archives-ouvertes.fr/hal-00894120

S. R. Searle, Density Estimation for Statistics and Data Analysis, 1982.

D. Sorensen, S. Andersen, D. Gianola, and I. Korsgaard, Bayesian inference in threshold models using Gibbs sampling, Genet Sel Evol, vol.27, pp.229-249, 1995.
URL : https://hal.archives-ouvertes.fr/hal-00894092

D. Sorensen, Gibbs sampling in quantitative genetics, Danish Institute of Animal Science, vol.82, 1996.

J. F. Taylor, B. Bean, C. E. Marshall, and J. J. Sullivan, Genetic and environmental components of semen production traits of artificial insemination Holstein bulls, J Dairy Sci, vol.68, pp.2703-2722, 1985.

R. J. Tempelman and D. Gianola, Marginal maximum likelihood estimation of variance components in Poisson mixed models using Laplacian integration, Genet Sel Evol, vol.25, pp.305-319, 1993.
URL : https://hal.archives-ouvertes.fr/hal-00893998

L. Tierney and J. B. Kadane, Accurate approximations for posterior moments and marginal densities, J Am Stat Assoc, vol.81, pp.82-86, 1986.

L. Tierney, R. E. Kass, and J. B. Kadane, Approximate marginal densities of non-linear functions, Biometrika, vol.76, pp.425-433, 1989.

L. Tierney, R. E. Kass, and J. B. Kadane, Approximate marginal densities of non-linear functions: Amendments and corrections, Biometrika, vol.78, pp.233-234, 1991.

C. P. Van-tassell, V. Vleck, and L. D. , Multiple-trait Gibbs sampler for animal models: Flexible programs for Bayesian and likelihood-based (co)variance component inference, J Anim Sci, vol.74, pp.2586-2597, 1996.

C. S. Wang, J. J. Rutledge, and D. Gianola, Bayesian analysis of mixed linear models via Gibbs sampling with an application to litter size in Iberian pigs, Genet Sel Evol, vol.26, pp.91-115, 1994.
URL : https://hal.archives-ouvertes.fr/hal-00894021

, APPENDIX: DERIVATIVES OF THE LOG POSTERIOR DENSITY ON THE CANONICAL SCALE