marginal posterior distribution Given $n$ normally distributed observations $f(X|\sigma_X)=\mathcal{N}(\mu_X, \sigma_X^2)$ and assuming a uniform prior on $\log(\sigma_X)$ and known $\mu_x$, I'm trying to find marginal posterior distribution for $\sigma^2_X$. I know how to approach for finding marginal posterior distribution for $\sigma_X$, but not $\sigma^2_X$ and even in that case, I don't find any nice form for the posterior, so I was not sure if my approach is correct. 
Here is my try:
$$P(\sigma_X|X)=P(X|\sigma_X)P(\sigma_X)$$
$$f(log(\sigma_X))\propto 1 \qquad \text{therefore, }\qquad f(\sigma_X)\propto\frac{1}{\sigma_X}$$
$$P(\sigma_X|X)\propto \frac{1}{\sigma_X}\prod_i^n\frac{1}{\sqrt{2\pi\sigma^2_X}}\exp(-\frac{(x_i-\mu_X)^2}{2\sigma_X^2})$$
Which seems to me that is no particular distribution form. My questions are:
1) Is my interpretation of the uniform prior on log scale correct?
2) Does this final form represent any particular distribution function?
3) Is there a way to find marginal posterior distribution for $\sigma^2_X$ or should I assume that this was a typo and we only can find marginal posterior distribution for $\sigma_X$?
 A: In the case where $\mu$ is known, there is no "marginal" posterior distribution, only a posterior distribution, and you already have it.  To see this, start out by changing your prior to be on $\sigma^2$; as it happens, making it uniform on $\log \sigma^2$ gives you $f(\sigma^2) \propto 1/\sigma$, as before.  (The answer to your first question is yes, you are interpreting the uniform prior on the log scale correctly.)  Your posterior is then $P(\sigma^2|X)$ with exactly the same functional form as you have already derived; perform the product multiplication from $1$ to $n$ to get:
$$P(\sigma^2|X,\mu) \propto (\sigma^2)^{-(n+1)/2}\exp\left(-{\sum(x_i-\mu)^2 \over 2\sigma^2}\right)$$
which is simply a rewritten version of what you already have.  
As for what distribution this is, if you look at the functional form, it looks a lot like that of a Gamma distribution, just with the variable of interest in the denominator everywhere instead of the numerator.  This leads us to the inverse gamma distribution (Wikipedia link.)
