I have quite the specific question over a model for which I am asked to compute a posterior. Here are all the details : Bayesian model .
For clarity purposes, let $X=[x_1^{\top}, \ldots, x_T^{\top}]^{\top}\in \mathbb R^{T\times k}$, $\varepsilon = [\varepsilon_1, \ldots, \varepsilon_T]^{\top}\in \mathbb R^{T}$, $Y=[y_1, \ldots, y_T]^{\top}\in \mathbb R^{T}$ and $U=[u_1^{\top}, \ldots, u_T^{\top}]^{\top}\in \mathbb R^{T\times l}$
My question revolves about the factorization of the joint posterior that you can see at the end of the second question. Thanks to the hint of my professor, I was able to progress for the first two factors:
By noting that $ Y-U\phi = X \beta +\varepsilon$, I found :
\begin{equation} \beta \mid (Y, X, U), \phi, \sigma^2 = \Sigma_XX^{\top}(Y-U\phi - \varepsilon)\sim \cal N( \Sigma_X(Y-U\phi), \sigma^2 \Sigma_X \Sigma_X^{\top}),\end{equation} where $\Sigma_X = (X^{\top}X)^{-1}$
I could then easily compute the posterior distribution through the Bayes formula: \begin{equation} \pi(\beta\mid (Y, X, U), \phi, \sigma^2, \gamma, q=1) \propto \pi(\beta \mid \sigma^2, \gamma, q=1) \times L(\beta \mid (Y, X, U), \phi, \sigma^2). \end{equation}
A similar manipulation led me to find the posterior for $\phi$, however I am failing to understand how to progress regarding the posterior for $\sigma^2$.
My intuition led me to do this: \begin{equation} X\beta + \varepsilon \mid ((Y,X,U),\gamma) \sim \cal N(0,\sigma^2(\gamma^2X^{\top}X+I_k). \end{equation} $\gamma^2X^{\top}X$ being definite positive, its eigenvalues are positive. So $-1<0$ is not an eigenvalue of $X^{\top}X$, which means $X^{\top}X+I_k$ is definite positive too, thus admits $S$ invertible so that $S^{\top}S=\gamma^2X^{\top}X+I_k$. We can then write that $S^{-1}(X\beta+\varepsilon) \mid ((Y,X,U),\gamma) \sim \cal N(0,\sigma^2 I_k)$.
This seems to be the most intuitive way to advance in this exercise, but from there on, I am not sure how I can continue.
My goal is to compute $\pi(\sigma^2\mid (Y,X,U),\gamma) \propto \pi(\sigma^2)\times L(\sigma^2\mid (Y,X,U),\gamma)$ .
My question is: how can I compute $L(\sigma^2\mid (Y,X,U),\gamma)$? Am I even heading the right way, for that regard?
Thank you for your answers, and apologies if my text is not very elegant, I am not well-versed in the art of writing math through a computer.