Solving a Very Challenging Question: Guidance Needed

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Finally, use Bayes' rule to obtain the posterior distribution for p(t|y). Based on this, we can see that the posterior distribution follows a Student's t-distribution with n degrees of freedom and a non-centrality parameter of (n+1)/c. This distribution is then used to derive the desired expression for p(t|y).In summary, the conversation discusses the use of p(\sigma^2) and p(\mu|\sigma^2) to calculate the posterior distribution p(t|y). It is suggested to break the problem into steps and use Bayes' rule to obtain the desired expression for p(t|y). This results in a Student's t-distribution with n degrees of freedom and a non
  • #1
gerv13
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Hi, can someone please help me just START this question or give me hints on what to do because i have no idea what to do:

[tex]Y_i| \mu, \sigma^2[/tex]~[tex]N(\mu,\sigma^2)[/tex]
use [tex] p(\sigma^2) \propto \frac{1}{\sigma^2} [/tex] and [tex]p(\mu|\sigma^2) = \frac{1}{\sqrt{2\pi}\sqrt{c}\sigma} exp[-\frac{1}{2} \frac {\mu^2}{c \sigma^2}][/tex]

and show that
[tex]p(t|y) \propto (1 + \frac{t^2}{n})^{-(\frac{n+1}{2})}[/tex]

where
[tex]
t = \frac{\sqrt{n + 1/c}}{\sqrt{\frac{s}{n} + \frac{\overline{y^2}(1/c)}{(n+1/c)}}} (\mu - \frac{n \overline{y}}{n + 1/c}) [/tex]

any guidance would be VERY appreciated because I've just been staring at this question for the past two days... Thank you?
 
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  • #2
Hint: Break the problem into steps. First, calculate the marginal likelihood for each of the parameters (mu and sigma^2). Then, use those results to calculate the likelihood for t given y.
 

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