Posterior mean of multivariate normal distribution in Murphy's Probabilistic Machine Learning In the new Probabilistic Machine Learning book by Kevin P. Murphy, to formulate the posterior mean of a multivariate normal distribution, he first defined the likelihood as

However, I can't seem to understand how to go from Equation 4.188 to 4.189 and subsequently to 4.190. Also, how can we simply replace the observations with their mean?
It would be an enormous help if somebody can explain it to me or perhaps guide me how to derive this. Thank you so much!

From the likelihood, where the normalizing constant should be $(2\pi)^{-N\cdot D/2}\begin{vmatrix}\!\Sigma\!\end{vmatrix}^{-N/2}$, we see that
$$\mathbf{Y}\sim\mathcal{N}_{D\cdot N}\left({\left(\mu,\ldots,\mu\right)}^\top,\mathrm{blockdiag}\left(\Sigma,\ldots,\Sigma\right)\right)$$
for the vector $\mathbf{Y}={(Y_1,\ldots,Y_N)}^\top$.
Thus, we have
$$\bar{Y}\sim\mathcal{N}_{D}\left(\mu,\frac{1}{N}\Sigma\right).$$
Considering the likelihood of $\bar{y}$ as function of $\mu$, we get
$$
\begin{align} 
\mathcal{L}\left(\mathbf{\mu};\bar{y}\right)&\overset{\mu}{\propto}\exp\left(-\frac{1}{2}\left(N\bar{y}^\top \Sigma^{-1}\bar{y}-2\mu^\top\Sigma^{-1}N\bar{y}+N\mu^\top\Sigma^{-1}\mu\right)\right)\\
&\overset{\mu}{\propto} \exp\left(-\frac{1}{2}\left(-2\mu^\top\Sigma^{-1}N\bar{y}+N\mu^\top\Sigma^{-1}\mu\right)\right)\\
&= \exp\left(-\frac{1}{2}\sum_{n=1}^N\left(-2\mu^\top\Sigma^{-1}y_n+\mu^\top\Sigma^{-1}\mu\right)\right)\\
&\overset{\mu}{\propto} \exp\left(-\frac{1}{2}\sum_{n=1}^N\left(y_n^\top\Sigma^{-1}y_n-2\mu^\top\Sigma^{-1}y_n+\mu^\top\Sigma^{-1}\mu\right)\right)\\
&\overset{\mu}{\propto} \mathcal{L}\left(\mathbf{\mu};\mathbf{y}\right).
\end{align}
$$
Maybe this is what the excerpt from the book is trying to convey.
