In RVM(relevant vector machine, also named Sparse Bayesian Learning), there are two important factors $s_i$ and $q_i$, the $q_i$ has been explained as 'quality factor'. In Bishop's PRML section 7.2.2 and many other alike materilas, it has $$ q_i = \boldsymbol{\phi_i^TC_{-i}^{-1}t} $$ and it describes $q_i$ as a measure of the alignment of $\boldsymbol{\phi_i}$ with the error of the model with that vector excluded. I can't understand why we can think of $q_i$ as the error meansurement from this formula, I think the result is not so intuitive. Aslo, I found more detailed information in paper <Fast Marginal Likelihood Maximisation for Sparse Bayesion Models>(http://www.miketipping.com/papers/met-fastsbl.pdf), in this paper it says:
...The 'quality factor' can be written as $q_i = \sigma^{-2}\boldsymbol{\phi_i^T(t-y_{-i})}$...
But there is no proof about how can we transform $\boldsymbol{\phi_i^TC_{-i}^{-1}t}$ to $\sigma^{-2}\boldsymbol{\phi_i^T(t-y_{-i})}$. Note that in some material use $\beta^{-1}$ instead of $\sigma^2$. So, anyone knows how to get $\sigma^{-2}\boldsymbol{\phi_i^T(t-y_{-i})}$ from $\boldsymbol{\phi_i^TC_{-i}^{-1}t}$ ? I know that $\boldsymbol{t=y +}\epsilon = \boldsymbol{\Phi w} + \epsilon= \boldsymbol{\Phi_{-i}w_{-i}+ \phi_i}w_i + \epsilon=\boldsymbol{y_{-i}+\phi_i}w_i + \epsilon$.