I am given the joint posterior pdf:

$$f(\mu,\sigma^2\mid\mathbf x) \propto (\sigma^2)^{-n/2-1}\exp{\left(-\frac{1}{2}\frac{\sum^n_{i=1}(x_i-\mu)^2}{\sigma^2}\right)}$$

I understand that marginal posterior pdfs can be obtained by integrating and removing out the other variable. With the given equation:

$$\sum^n_{i=1}(x_i-\mu)^2 = \sum^n_{i=1}(x_i-\bar x)+n(\mu-\bar x)^2$$

I tried to show that the marginal pdf $f(\sigma^2|\mathbf x)$ is proportional to

$$(\sigma^2)^{-n/2-1/2}\exp{\left(-\frac{1}{2}\frac{\sum^n_{i=1}(x_i-\bar x)^2}{\sigma^2}\right)}$$

however by substituting the given equation to the joint posterior pdf and integrating it with respects to $\mu$, I was still not able to obtain the correct marginal pdf $f(\sigma^2\mid\mathbf x)$.

Can anyone help me out please? Thank you.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.