Given is a linear model $y_i=x_i^T+e_i$ where $e_i \sim N(0, \sigma^2)$. The precision is defined as $\tau = \frac{1}{\sigma^2}$. I am asked to derive the posterior $\pi(\beta|x,\tau)$.
Would I compute the posterior by doing the following: $\pi(\beta|x,\tau) \propto p(x, \tau | \beta) p(\beta)$ or $\pi(\beta|x,\tau) \propto p(x,|\tau, \beta) p(\tau |\beta) p(\beta)$?
I would assume the above relations to be equivalent; How do the mathematical expressions of the pdfs of $p(x, \tau | \beta)$ and $p(x,|\tau, \beta$) differ?
