Noise precision updates

Here we give the update equation for the parameters of the noise precision distribution $q(\phi_{\epsilon_i}\vert y)=Ga(b_{\epsilon_i},c_{\epsilon_i})$:

$$\frac{1}{b_{\epsilon_i}} = \frac{1}{2} \left( \sum_t( y_{it}-x_tQ\mu_{B_i}-\sum_p(y_{i(t-p)}... ..._{B_i})a_{pi} )^2 + \text{Trace}(\Lambda_{\beta_i}^{-1}\sum_t x_t^Tx_t) \right.$$

$$+ \left. \sum_p \left\{ (\Lambda_{a_{p}}^{-1})_i\sum_t(y_{i(t-p)}-x_{t-p}Q\mu_{B_i})^2 \right\} \right) + \frac{1}{b_{\epsilon_0}}$$

$$c_{\epsilon_i} = (T-1)/2+c_{\epsilon_0}$$ (47)

we define $\gamma_{\epsilon_i}$ as:

$$\gamma_{\epsilon_i} = \frac{\Gamma(b_{\epsilon_i})\Gamma(c_{\epsilon_i}+1)}{\Gamma(c_{\epsilon_i})}$$ (48)