# 1-D posterior on higher dimensional iid data

This may have been asked before, in which case, apologies.

Say I know my posterior density $\pi(\theta|x)$ where $\theta,\,x\in \mathbb{R}$ and now have $iid$ data $\mathbf{x}=(x_{i})_{i=1}^{N}$, is the following appropriate for $\pi(\theta|\textbf{x})$: $$\pi(\theta|\textbf{x}) = \frac{\pi(\theta)\Pi_{i=1}^{N}p(x_{i}|\theta)}{\Pi_{i=1}^{N}p(x_{i})} \quad \quad \text{(Bayes and independence)} \\ = \frac{\pi(\theta)\Pi_{i=1}^{N}\left(\frac{\pi(\theta|x_{i})p(x_i)}{\pi(\theta)}\right)}{\Pi_{i=1}^{N}p(x_{i})} \quad \quad \text{(Bayes on }p(x_i|\theta)) \\ =\pi(\theta)^{-(n-1)}\Pi_{i=1}^{N}\pi(\theta|x_i).$$ Is this correct? Basically I want $\pi(\theta|\textbf{x})$ from $\pi(\theta|x)$ and the above is my current best guess.