get_equi_bayes: Get the Bayes rule under multiway Stein's loss.
In tensr: Covariance Inference and Decompositions for Tensor Datasets

Description Usage Arguments Details Value Author(s) References See Also Examples

Given the output of equi_mcmc, this function will calculate the Bayes rule under multiway Stein's loss.

1	get_equi_bayes(psi_inv, sigma, burnin = NULL)

`psi_inv`	A list of arrays where `psi_inv[[i]][[, , j]]` is the jth update of the `i`th component. These components are the inverses of the lower-triangular Cholesky square roots of the component covariance matrices. You can just use the `Phi_inv` output from `equi_mcmc`.
`sigma`	A vector of posteior draws of the total variation parameter. This is just `sigma` from the output of `equi_mcmc`.
`burnin`	A numeric between 0 and 1. What proportion of the posterior samples do you want to discard as burnin? The default is 0.25.

Multiway Stein's loss is a generalization of Stein's loss to more than two dimensions. The Bayes rule under this loss is simply represented in terms of the posterior moments of the component precision matrices. These moments can be approximated by using the output of equi_mcmc. When using the invariant prior that is used in equi_mcmc, the resulting Bayes rule is the uniformly minimum risk equivariant estimator.

More details on multiway Stein's loss and the Bayes rules under it can be found in Gerard and Hoff (2015).

Sig_hat A list of the Bayes rules of the component covariance matrices under multiway Stein's loss.

B A list of the lower-triangular Cholesky square roots of the Bayes rules of the component covariance matrices under multiway Stein's loss. We have that Sig_hat[[i]] is equal to B[[i]] %*% t(B[[i]]).

b A numeric. This is the bayes rule of the total variation parameter. This is the 'standard deviation' version. That is, the b ^ 2 would be used to calculate the overall covariance matrix.

David Gerard.

Gerard, D., & Hoff, P. (2015). Equivariant minimax dominators of the MLE in the array normal model. Journal of Multivariate Analysis, 137, 32-49. https://doi.org/10.1016/j.jmva.2015.01.020 http://arxiv.org/pdf/1408.0424.pdf

equi_mcmc.

#Generate data whose true covariance is just the identity.
p <- c(4,4,4)
X <- array(stats::rnorm(prod(p)),dim = p)
#Then run the Gibbs sampler.
mcmc_out <- equi_mcmc(X)
bayes_rules <- get_equi_bayes(mcmc_out$Phi_inv, mcmc_out$sigma)
bayes_rules$Sig_hat[[1]]