The equation below is a posteriori density under the assumption that the prior distribution for $\mu$ is normal.
And I wonder why do the highlighted portions where $\mu$ and $x_{k}$ are swapped when expressing the multiplication of exponential quotients into summation. Can anyone please explain or is it the mistake?
Thank you for your help.
the aim to swap $(x_k-\mu)^2=(\mu-x_k)^2$ is to show that the posterior density (now the rv is $\mu$, because $x_i$ are known data) is still gaussian.
Next step is to observe that
$$\Sigma_k(\mu-x_k)^2=n(\mu-\overline{x})^2+C$$
where $C$ is an expression without $\mu$ so it can be canceled and included in the normalizing constant $\alpha$... and so on