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Math Help - posterior

  1. #1
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    posterior

    For this question,
    for  Y_i|\mu,\sigma^2 ~ N(\mu,\sigma^2)
    use  p(\sigma^2) \propto \frac{1}{\sigma^2} and  p(\mu|\sigma^2) = \frac{1}{\sqrt{2\pi}\sqrt{c}\sigma} exp[-\frac{1}{2} \frac {\mu^2}{c \sigma^2}] i.e.  \mu|\sigma^2 ~ N (0,c\sigma^2)
    and show  p(\mu,\sigma^2|y) \propto (\sigma^2)^{-(\frac{n+1}{2})-1} exp [-\frac{1}{\sigma^2}(\frac{s+n(\mu - \overline{y})^2 + \frac{\mu^2}{c}}{2})]

    can you please check that Im on the right track:
     p(\mu, \sigma^2|y) \propto p(y|\mu, \sigma^2)p(\mu|\sigma^2)p(\sigma^2)
     \propto \frac{1}{\sqrt{2\pi}\sqrt{c}\sigma} exp[-\frac{1}{2} \frac {\mu^2}{c \sigma^2}] \times \frac{1}{\sigma^2} \times \prod(\sigma^2)^{-1/2} exp [-1/2 \frac{(Y_i - \mu)^2}{\sigma^2}]

    any help would be very appreciated coz i have a STRONG feeling im somehow very off track even though im still at the start of the question.. thanks
    Last edited by bubbling; September 15th 2009 at 04:08 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by bubbling View Post
    For this question,
    for  Y_i|\mu,\sigma^2 ~ N(\mu,\sigma^2)
    use  p(\sigma^2) \propto \frac{1}{\sigma^2} and  p(\mu|\sigma^2) = \frac{1}{\sqrt{2\pi}\sqrt{c}\sigma} exp[-\frac{1}{2} \frac {\mu^2}{c \sigma^2}] i.e.  \mu|\sigma^2 ~ N (0,c\sigma^2)
    and show  p(\mu,\sigma^2|y) \propto (\sigma^2)^{-(\frac{n+1}{2})-1} exp [-\frac{1}{\sigma^2}(\frac{s+n(\mu - \overline{y})^2 + \frac{\mu^2}{c}}{2})]

    can you please check that Im on the right track:
     p(\mu, \sigma^2|y) \propto p(y|\mu, \sigma^2)p(\mu|\sigma^2)p(\sigma^2)
     \propto \frac{1}{\sqrt{2\pi}\sqrt{c}\sigma} exp[-\frac{1}{2} \frac {\mu^2}{c \sigma^2}] \times \frac{1}{\sigma^2} \times \prod(\sigma^2)^{-1/2} exp [-1/2 \frac{(Y_i - \mu)^2}{\sigma^2}]

    any help would be very appreciated coz i have a STRONG feeling im somehow very off track even though im still at the start of the question.. thanks
    You have omitted that the Y_i s are independent. You can also lost some more costant multiplers, but on the whole that looks OK (Oh.. in your last expression the Y_i 's should be y_i s)

    CB
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  3. #3
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    Thanks for the help, I've finished that question now . But can you please help me start this question because i have no idea how to start (it's similar to the first i think):

    for  Y_i|\mu,\sigma^2 ~ N(\mu,\sigma^2)

    use  p(\sigma^2) \propto \frac{1}{\sigma^2} and  p(\mu|\sigma^2) = \frac{1}{\sqrt{2\pi}\sqrt{c}\sigma} exp[-\frac{1}{2} \frac {\mu^2}{c \sigma^2}] i.e.  \mu|\sigma^2 ~ N (0,c\sigma^2)

    show that

    <br /> <br />
p(t|y) \propto (1 + \frac{t^2}{n})^{-(\frac{n+1}{2})}

    where

    <br /> <br />
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})<br /> <br />
    Last edited by bubbling; September 13th 2009 at 04:55 AM.
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