Given a sample X of size n from a normal distribution $N(\mu,\sigma)$ one can estimate $\sigma$ by $\hat{\sigma}$ from X. Then we know that:

$p(\sigma|\hat{\sigma},n)\propto\hat{\sigma}\sqrt{\frac{n-1}{\chi^2(\nu=n-1)}}$

Now I am looking for the likelihood function: $p(\hat{\sigma}|\sigma,n)$.

Is there an analytic solution for this or is it necessary to derive the likelihood function from case-by-case simulations?

Can someone help me with this relation? Or better, point to a reference where this relation has been used before?