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?

**Answer**

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.

$$

**Attribution***Source : Link , Question Author : Nussig , Answer Author : A. Donda*