Joint Maximization of Log-Likelihood and Two Parameters that Decouple from each other In the first chapter of Bishop's book, section 1.2.4 (Gaussian Distribution), he is trying to determine the parameters of a Gaussian distribution given a data-set $x = (x_{1}, ..., x_{n})$. He assumes i.i.d. data-points that are drawn from this Gaussian distribution. He first takes the log of

and reaches to

At first, by maximizing the log likelihood with respect to $\mu$ he finds

Next, he determines the variance by maximizing the log likelihood with respect to  $\sigma^{2}$

Then he explains that they are performing "joint maximization" of the log Likelihood, "but
in the case of the Gaussian distribution the solution for $\mu$ decouples from that for $\sigma^{2}$
so that we can first evaluate $\mu_{ML}$ and then subsequently use this result to evaluate $\sigma^{2}_{ML}$."
What I would like to learn about is that how this joint maximization is done and what does he mean by mentioning that $\mu$ and $\sigma^{2}$ decouple from each other?

In general
$$\max_{\mu,\sigma}\ell(\mu,\sigma)=\max_{\sigma}\max_{\mu}\ell(\mu,\sigma)=\max_{\sigma}\ell(\hat\mu(\sigma),\sigma)$$
where
$$\hat\mu(\sigma)=\arg\max_{\mu}\ell(\mu,\sigma)$$
but in the event $\hat\mu(\sigma)$ is independent of $\sigma$
$$\max_{\mu,\sigma}\ell(\mu,\sigma)=\max_{\sigma}\ell(\hat\mu,\sigma)$$
