Prior distributions of the Bayesian AMMI model

bam_priors() sets the hyperparameters of the prior distributions of bayes_ammi(). The defaults are the vague but proper priors of Crossa et al. (2011): normal priors with mean zero and variance $10^8$ for the overall mean, the main effects and the singular values, and a scaled inverse chi-square prior with one degree of freedom for the error variance.

Usage

bam_priors(
  mu_mean = 0,
  mu_var = 1e+08,
  tau_mean = 0,
  tau_var = 1e+08,
  delta_mean = 0,
  delta_var = 1e+08,
  lambda_mean = 0,
  lambda_var = 1e+08,
  nu = 1,
  s2 = NULL
)

# S3 method for class 'bam_priors'
print(x, ...)

Arguments

mu_mean, mu_var: prior mean and variance of the overall mean $\mu$.
tau_mean, tau_var: prior mean (a scalar or a vector of length $G$) and variance of the genotype effects $\tau_i$.
delta_mean, delta_var: prior mean (scalar or length $E$) and variance of the environment effects $\delta_j$.
lambda_mean, lambda_var: prior mean and variance of the singular values $\lambda_k$.
nu, s2: degrees of freedom $\nu$ and scale $S^2_0$ of the prior of the error variance (s2 = NULL: the least-squares residual mean square).
x: a "bam_priors" object.
...: unused.

Value

A list of class "bam_priors" with these elements.

Details

With $G$ genotypes, $E$ environments and $K$ bilinear terms, $$\mu \sim N(\mu_0, \sigma^2_\mu), \quad \boldsymbol{\tau} \sim N(\mathbf{m}_\tau, \sigma^2_\tau \mathbf{I}_G), \quad \boldsymbol{\delta} \sim N(\mathbf{m}_\delta, \sigma^2_\delta \mathbf{I}_E),$$ each main effect restricted to sum zero, $$\lambda_k \sim N^{+}(\lambda_0, \sigma^2_\lambda), \quad \lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_K > 0, \qquad \sigma^2 \sim \mathrm{Inv}\textrm{-}\chi^2(\nu, S^2_0),$$ and spherical uniform priors for the singular vectors on the subspaces orthogonal to $\mathbf{1}$ and to the other vectors. $S^2_0$ = NULL (default) takes the residual mean square of the least-squares AMMI fit, so that the prior is equally vague whatever the unit of the trait; with $\nu = 1$ it carries the information of a single observation. Crossa et al. (2011) report $S_0 = 10^8$; their published posterior of $\sigma^2$ is reproduced with the vague default (see the vignette on validation).

References

Crossa, J., Perez-Elizalde, S., Jarquin, D., Cotes, J. M., Viele, K., Liu, G. and Cornelius, P. L. (2011). Bayesian estimation of the additive main effects and multiplicative interaction model. Crop Science, 51(4), 1458–1469. doi:10.2135/cropsci2010.06.0343

Hobert, J. P. and Casella, G. (1996). The effect of improper priors on Gibbs sampling in hierarchical linear mixed models. Journal of the American Statistical Association, 91(436), 1461–1473.

Author

Muhammad Yaseen (myaseen208@gmail.com)

Examples

bam_priors(
    mu_mean     = 0
  , mu_var      = 1e8
  , tau_mean    = 0
  , tau_var     = 1e8
  , delta_mean  = 0
  , delta_var   = 1e8
  , lambda_mean = 0
  , lambda_var  = 1e8
  , nu          = 1
  , s2          = NULL
  )
#> Priors of the Bayesian AMMI model
#>   mu            ~ N(0e+00, 1e+08)
#>   tau_i         ~ N(0e+00, 1e+08), sum zero
#>   delta_j       ~ N(0e+00, 1e+08), sum zero
#>   lambda_k      ~ N+(0e+00, 1e+08), ordered
#>   sigma^2       ~ Inv-chi^2(nu = 1e+00, s2 = least-squares residual mean square)
#>   alpha_k, gamma_k uniform on the constrained unit spheres