confusion with this derivation

Question

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

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

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

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

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.

Answer 1

As provided, the answer to the question is that you are correct. The expression$$\frac{1}{\sigma^2}\sum_{i=1}^{N} <w_i> y_ix_i$$ is certainly inappropriate as definitely not equal to the expectation of$$\sum_{i=1}^{N} \frac{w_i}{\sigma^2} (y_i - \beta x_i)^T(y_i - \beta x_i)$$It would be nice though if you could provide an excerpt of the paper you refer to, so that we can check this is indeed what the author(s) meant.