next up previous
Next: Autoregressive parameter updates Up: Variational Bayes Updates Previous: Variational Bayes Updates

Regression coefficient updates

Here we give the update equation for the parameters of the regression coefficient distribution $ q(\beta_i,\bar{\beta}_i\vert y)=MVN(\mu_{B_i},\Lambda_{B_i})$:

$\displaystyle \mu_{B_i}$ $\displaystyle =$ $\displaystyle F_{B_i}^{-1}E_{B_i}$  
$\displaystyle \Lambda_{B_i}$ $\displaystyle =$ $\displaystyle F_{B_i}^{-1}$ (39)

where:
$\displaystyle F_{B_i}$ $\displaystyle =$ $\displaystyle \gamma_{{\epsilon_i}} \sum_t
\left\{Q^Tx_{i}^Tx_tQ+\sum_p{\mu_{a_{pi}}(Q^Tx_{t-p}^Tx_{t-p}Q-2Q^Tx_t^Tx_{t-p}Q)}
\right\}$  
    $\displaystyle +\sum_e
\left\{\phi_{\bar{\beta}_e}^{(i)}(Q_e^TCQ_e+R_e^Tm^TCmR_e-2Q_e^TCmR_e)\right\}$  
$\displaystyle E_{B_i}$ $\displaystyle =$ $\displaystyle \gamma_{{\epsilon_i}} \sum_t \left\{
Q^Tx_t^T(y_{it}-\sum_p{y_{i(t-p)}\mu_{a_{pi}}})\right.$  
  $\displaystyle +$ $\displaystyle \left.\sum_p{\mu_{a_{pi}}Q^Tx_{t-p}^T(y_{it}-\sum_p{y_{i(t-p)}\mu_{a_{pi}}})}
\right\}$  
$\displaystyle \gamma_{{\epsilon_i}}$ $\displaystyle =$ $\displaystyle \frac{\Gamma(b_{{\epsilon_i}})\Gamma(c_{{\epsilon_i}}+1)}{\Gamma(c_{{\epsilon_i}})}$ (40)