Concepts

Notations

Notation	Definition	Notes
$J$	Number of response variables
$N$	Sample size
$Y\mapsto{\rm R}^{J}$
$X$
$\beta\mapsto{\rm R}^{J}$
$\Sigma\mapsto{\rm R}^{J \times J}$
$V$

Basic association model

Consider a multivariate regression problem $$Y \mid X, \beta, \Sigma \sim N_J(X\beta, \Sigma)$$ The goal is to make inference on $\beta$. A Bayesian model for $\beta$ is adopted $$\beta \mid U \sim N_J(0, U)$$

A Bayesian hierachical model with a spike-slap prior on $\beta$ is adopted
\[\beta \mid \bar{\beta}, U_\phi, \pi_0 \sim \pi_0\delta_0 + (1 - \pi_0)N_J(\bar{\beta}, U_\phi)\]
\[\bar{\beta} \mid U_\omega \sim N_J(0, U_\omega)\]
Therefore, \[\beta \mid U_\phi, U_\omega, \pi_0 \sim \pi_0\delta_o + (1 - \pi_0) N_J(0, U_\phi + U_\omega)\]
Let $U = U_\phi + U_\omega$, then \[\beta \mid U, \pi_0 \sim \pi_0\delta_0 + (1 - \pi_0)N_J(0, U)\]