Posterior distribution of a parameter, conditional on another parameter I am trying  to derive conditional posterior distributions  of parameters $\theta$ and $\sigma^2$. I know that $X_1,...,X_n$ are i.d.d. $N(\theta,\sigma^2)$ and joint  probability distribution for $\theta$ and $\sigma^2$ is:
$$\pi(\theta,\sigma^2) \propto \frac{1}{\sigma^2}, \theta \in (-\infty,\infty), \sigma^2 \in (0,\infty)$$
I need to derive $\pi(\theta\mid\sigma^2,\textbf{x})$  and $\pi(\sigma\mid\theta,\textbf{x})$. I managed to derive joint posterior probability $\pi(\theta,\sigma^2\mid \textbf{x})$ but after that I am pretty much stuck. Is it possible to derive what I need based on existing information?

The conditional densities can be directly derived from the joint density:
\begin{align}
\pi(\theta\mid\sigma^2,\textbf{x}) &\propto \pi(\theta,\sigma^2\mid\textbf{x})\\
\pi(\sigma^2\mid\theta,\textbf{x}) &\propto \pi(\theta,\sigma^2\mid\textbf{x})
\end{align}
where the proportionality sign ($\propto$) is meant as a function of $\theta$ in the first case and as a function of $\sigma^2$ in the second case. Left- and right-hand terms are identical functions of $\theta$ and $\sigma^2$, respectively, up to a multiplicative constant that may depend on $\mathbf x$ and on $\sigma^2$ in the first case and on $\theta$ in the second case.
