Well I have struggled with this for some days now. Let be $n$ observations $y=(y_1,y_2,...,y_n)$, where $y_i|\mu,\sigma^2 \sim \mathcal{N}(\mu,\sigma^2),1\leq i\leq n$ assumed to be conditionally independent given and both parameters and are unknown.
I have managed to find Jeffreys prio, I am struggling calculating a prior for this likelyhood using the exponential family method.
What i have done so far \begin{equation*} P(y|\mu,\sigma^2) \overset{(4)}{=} \frac{1}{(2\pi\sigma^2)^{\frac{n}{2}}} \exp{\Big{\{}{\frac{n\mu^2}{2\sigma^2}}\Big{\}} \exp{\Big{\{}-}\frac{\sum_{i=0}^{n}(y_{i}^2-2y_i\mu)}{2\sigma^2}\Big{\}}} \\= \frac{1}{(2\pi\sigma^2)^{\frac{n}{2}}} \exp{\Big{\{}{\frac{n\mu^2}{2\sigma^2}}\Big{\}}} \exp{\Big{\{}}\frac{-\sum_{i=0}^{n}y_{i}^2}{2\sigma^2}+\frac{\sum_{i=0}^{n}y_i}{2\sigma^2}\Big{\}}\\ \end{equation*} It belongs to the exponential family of distributions, specifically: \begin{equation*} g(\mu,\sigma^2) = \frac{1}{(2\pi\sigma^2)^{\frac{n}{2}}} \exp{\Big{\{}{\frac{n\mu^2}{2\sigma^2}}\Big{\}}} \hspace{0.5cm} ,t(x) = (\sum_{i=0}^{n}y_{i}^2,\sum_{i=0}^{n}y_i)^T \hspace{0.5cm} ,c(\mu,\sigma^2) =(-\frac{1}{2\sigma^2},\frac{2\mu}{2\sigma^2}) \end{equation*}
And so I get
$P(\theta)=\frac{1}{(2\pi\sigma^2)^{\frac{n}{2}}} \exp{\Big{\{}{\frac{n\mu^2}{2\sigma^2}}\Big{\}}} \exp{\Big{\{}}\frac{-b_1}{2\sigma^2}+\frac{2\mu b_2}{2\sigma^2}\Big{\}}\\$
Even though I did not conclude a familiar distribution from this prior I decided to move on and calculate the marginal posteriors as it is $P(\mu|y) \propto \int P(\mu,\sigma^2|y)d\sigma^2,\hspace{0.8cm} P(\sigma^2|y) \propto \int P(\mu,\sigma^2|y)d\mu$
Well, lots of pencils and effort later I have some random distribution, which I am also not familiar and even though some things canceled out in the way reducing the noise I still can't get a proper result. I got some insight from https://en.wikipedia.org/wiki/Conjugate_prior#cite_note-posterior-hyperparameters-3 http://www.stats.org.uk/priors/noninformative/YangBerger1998.pdf and tried forming Inverse-gamma distribution(which was a failure), but I strongly believe I am missing something, specificaly in the parametrization of the exponential family form (should I go for more parameters?).
P.S. If the question is not explanatory enough I can post every step I did so far.
Well, after abandoning the exponential family method for determining a prior, I concluded this one using the normal inverse chi squared distribution after the following parametrization. I found the joint posterior as well. I am now trying to numericaly validate my results with R.
