Jeffreys Prior for normal distribution with unknown mean and variance I am reading up on prior distributions and I calculated Jeffreys prior for a sample of normally distributed random variables with unknown mean and unknown variance.
According to my calculations, the following holds for Jeffreys prior:
$$ p(\mu,\sigma^2)=\sqrt{\det(I)}=\sqrt{\det\begin{pmatrix}1/\sigma^2 & 0 \\ 0 & 1/(2\sigma^4)\end{pmatrix}}=\sqrt{\frac{1}{2\sigma^6}}\propto\frac{1}{\sigma^3}.$$
Here, $I$ is Fisher's information matrix.
However, I have also read publications and documents which state

*

*$p(\mu,\sigma^2)\propto 1/\sigma^2$ see Section 2.2 in Kass and Wassermann (1996).

*$p(\mu,\sigma^2)\propto 1/\sigma^4$ see page 25 in Yang and Berger (1998)
as Jeffreys prior for the case of a normal distribution with unkown mean and variance.
What is the 'actual' Jeffreys prior?
 A: The existing answers already well answer the original question. As a physicist, I would just like to add to this discussion a dimensionality argument. If you consider $\mu$ and $\sigma^2$ to describe a distribution of a random variable in a real 1D space and measured in meters, they have the dimensions $[\mu] \sim m$ and $[\sigma^2] \sim m^2$. To have a physically correct prior, you need it to have the right dimensions, i.e. the only powers of $\sigma$ physically possible in a non-parametric prior are: 
$$
\pi(\mu, \sigma) \sim 1/\sigma^{2}
$$ 
and 
$$
\pi(\mu, \sigma^2) \sim 1/\sigma^{3}
$$.
A: $\frac{1}{\sigma^3}$ is the Jeffreys prior. However in practice $\frac{1}{\sigma^2}$ is quite often used cause it leads to a relatively simple posterior, the "intuition" of this prior is that it corresponds with a flat prior on $\log(\sigma)$.
A: I think the discrepancy is explained by whether the authors consider the density over $\sigma$ or the density over $\sigma^2$. Supporting this interpretation, the exact thing that Kass and Wassermann write is
$$
\pi(\mu, \sigma) = 1 / \sigma^2,
$$
while Yang and Berger write
$$
\pi(\mu, \sigma^2) = 1 / \sigma^4.
$$
