# Confidence Interval vs Credible Interval for the Variance

I understand the conceptual difference between confidence and credible intervals. But I have difficulties applying these concepts to my application.

I would like to know the concrete difference between the confidence interval and the credible interval for the variance $\sigma^2$ . For simplicity, we can assume normally distributed data.

The calculation of the confidence interval is well known (e.g., 1): Given a normal distribution with a fixed $\sigma^2$, the sample variance $s^2$ has a chi-squared distribution. Based on this knowledge, one can derive the quantiles for $s^2$ and subsequently, by solving the resulting inequalities for $\sigma^2$, one can derive the confidence interval for $\sigma^2$. (The sampling distribution of $\sigma^2$ is not a chi-squared distribution due to the transformation!)

I did not find any information on how to do it with credible intervals. For this, I assume I have to solve this equation:

$P(\sigma^2|s^2)=\frac{P(s^2|\sigma^2)P(\sigma^2)}{P(s^2)}$.

Intuitively, I would argue that the likelihood $P(s^2|\sigma^2)$ has a chi-square distribution as well. Assuming flat priors $P(\sigma^2)$ and $P(s^2)$, does this ultimately mean that $P(\sigma^2|s^2)$ has a chi-squared distribution? In this case, the confidence interval and the credible interval for the variance are different, and not the same thing that is differently interpreted (which is the case for the interval estimates for the mean)?

As you can see, I completely base my "derivation" of the credible interval on intuition. It would be great if someone can provide a more rigorous treatment.