I am working through some inference maths as described in a paper and have a doubt about a certain step. At one point, the authors have to compute an expectation of the following expression

$$ \sum_{i=1}^{N} \frac{w_i}{\sigma^2} (y_i - \beta x)^T(y_i - \beta x) $$

Bow the expectation has to be taken wrt to $Q(w_i)$. For me, this should be simply:

$$ \frac{1}{\sigma^2}\sum_{i=1}^{N} <w_i> (y_i - \beta x)^T(y_i - \beta x) $$

where $<w_i>$ is the expectation operator applied. However, the authors write this as: $$ \frac{1}{\sigma^2}\sum_{i=1}^{N} <w_i> y_ix_i $$

I was wondering if there was some trick that I missed.