Suppose we're given a sequence $x_1,\ldots,x_n$ of realizations of i.i.d. $\mathcal{N}(\mu,\sigma^2)$ random variables and we want to apply maximum posterior estimation to estimate the parameters $\mu$ and $\sigma$. For this, suppose we decide $\mu$ and $\sigma$ have joint pdf $g = g(\mu,\sigma)$ on some set $A$. Then we seek to maximize $$ \frac{g(\mu,\sigma) \cdot \prod_{i=1}^n \phi(x_i \mid \mu,\sigma)}{\int_A g(x,y) \cdot \prod_{i=1}^n \phi(x_i \mid x,y)\, dx \, dy} $$ where $\phi(\cdot \mid \mu,\sigma) $ is the normal pdf.
The denominator, which is the marginal pdf of the data is dropped since it doesn't affect the maximum. My question is, how am I supposed to interpret the denominator as the marginal pdf of the data? I already assumed the observations were jointly normal, so it seems like the marginal pdf should be $$ \prod_{i=1}^n \phi(x_i ; \mu^*,\sigma^*) $$ for some parameters $\mu^*$ and $\sigma^*$. Is it that, I'm really forcing the marginal to be the denominator, whatever this integral works out to be? That still clashes with my understanding that the data is normal to begin with.