Edit since the question was closed due to asking more than one question: The question is how to get from $\eqref{1}$ $\eqref{2}$ $\eqref{3}$ to $\eqref{4}$ $\eqref{5}$ $\eqref{6}$ I think all the details are there so if something is missing or unclear please write it in the comments so I can try and fix it before voting it to be closed
Context:
$\bar{\Theta}$ is the room regression filter coefficients (RRC);
$$X_{t} = \bar{\Theta}^{H}\bar{X}_{t-1} + s_{t}$$ means in words: the filter that defines how the room causes reverberation to the signal (how the past of the speech signal x affects the current speech signal).
$s$ is drawn from $\mathcal{N}(0,\lambda_{t}).$
The goal of this analysis is to find the most likely $\bar{\Theta}$ with $\eqref{1}$ being the likelihood function.
I’m trying to get from these three expressions:
\begin{align} \mathcal{P}(\Theta | \bar{X}_{t}) &= \dfrac{\mathcal{P}(X_t|\Theta,\bar{X}_{t-1})\mathcal{P}(\Theta|\bar{X}_{t-1})}{\int{\mathcal{P}(X_{t}|\Theta,\bar{X}_{t-1})\mathcal{P}(\Theta|\bar{X}_{t-1})}}&&\text{due to Bayes}\tag{1}\label{1}\\ \mathcal{P}(X_{t}|\Theta,\bar{X}_{t-1}) &= \mathcal{N}(x_{t};\bar{\Theta}^H\bar{X}_{t-1},\lambda_{t})\tag{2}\label{2}\\ \mathcal{P}(\Theta|\bar{X}_{t-1}) &= \mathcal{N}(\bar{\Theta};\bar{\mu}_{t-1},\Phi_{t-1})\tag{3}\label{3} \end{align}
To the following expressions describing $\mathcal{P}(\Theta|\bar{X}_{t})$:
\begin{align} \mathcal{P}\left(\Theta|\bar{X}_{t}\right) &= \mathcal{N}\left(\bar{\Theta};\bar{\mu}_{t},\Phi_{t}\right)&\text{ with}\tag{4}\label{4}\\ \mu_{t} &= \Phi_{t}\left(\frac{\bar{X}_{t-1}X_{t}^{*}}{\lambda_{t}} + \Phi_{t-1}^{-1}\bar{\mu}_{t-1}\right)\tag{5}\label{5}\\ \Phi_{t} &= \left(\dfrac{\bar{X}_{t-1}X_{t-1}^{H}}{\lambda_{t-1}} + \Phi_{t-1}^{-1}\right)^{-1}\tag{6}\label{6} \end{align}
Things I tried: I think the denominator in $\eqref{1}$ is always 1 so there is no need to calculate that integral. I didn’t really get anywhere by just substituting the appropriate terms in $\eqref{1}$ with $\eqref{2}$ and $\eqref{3}$, so I thought since the goal is to find the most likely $\Theta$, $\eqref{5}$ & $\eqref{6}$ are the solution of the optimization problem, $\arg\max\limits_{\mu_{t-1},\Phi_{t-1}} \eqref{1}$.
Perhaps these expressions were derived from the appropriate gradients, and I got a similar expression for $\mu$ (not the same expression though) and nothing close with $\Phi$.