Bayesian Statistics: Marginal Posterior

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In a hierarchical model, the prior $pi(thetamidxi)$ for $theta$ depends on hyperparameters $xi$.



In my lecture notes, the following is now given:



$$ pi(xi mid x) = pi(theta,ximid x)over pi(thetamid x,xi).$$



I don't see how to derive this formula.




statistics bayesian density-function marginal-distribution




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  • Note that you can get nicer spacing around the vertical bars in conditional probabilities by using mid instead of |.
    – joriki
    Jul 17 at 18:56







  • 1




    oh wow! didn't know this, it indeed looks so much better this way! thank you :)
    – Protawn
    Jul 17 at 18:59














up vote
1
down vote

favorite












In a hierarchical model, the prior $pi(thetamidxi)$ for $theta$ depends on hyperparameters $xi$.



In my lecture notes, the following is now given:



$$ pi(xi mid x) = pi(theta,ximid x)over pi(thetamid x,xi).$$



I don't see how to derive this formula.




statistics bayesian density-function marginal-distribution




share|cite|improve this question





















  • Note that you can get nicer spacing around the vertical bars in conditional probabilities by using mid instead of |.
    – joriki
    Jul 17 at 18:56







  • 1




    oh wow! didn't know this, it indeed looks so much better this way! thank you :)
    – Protawn
    Jul 17 at 18:59












up vote
1
down vote

favorite









up vote
1
down vote

favorite











In a hierarchical model, the prior $pi(thetamidxi)$ for $theta$ depends on hyperparameters $xi$.



In my lecture notes, the following is now given:



$$ pi(xi mid x) = pi(theta,ximid x)over pi(thetamid x,xi).$$



I don't see how to derive this formula.




statistics bayesian density-function marginal-distribution




share|cite|improve this question













In a hierarchical model, the prior $pi(thetamidxi)$ for $theta$ depends on hyperparameters $xi$.



In my lecture notes, the following is now given:



$$ pi(xi mid x) = pi(theta,ximid x)over pi(thetamid x,xi).$$



I don't see how to derive this formula.





statistics bayesian density-function marginal-distribution





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 17 at 18:58
























asked Jul 17 at 18:25









Protawn

32129




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  • Note that you can get nicer spacing around the vertical bars in conditional probabilities by using mid instead of |.
    – joriki
    Jul 17 at 18:56







  • 1




    oh wow! didn't know this, it indeed looks so much better this way! thank you :)
    – Protawn
    Jul 17 at 18:59
















  • Note that you can get nicer spacing around the vertical bars in conditional probabilities by using mid instead of |.
    – joriki
    Jul 17 at 18:56







  • 1




    oh wow! didn't know this, it indeed looks so much better this way! thank you :)
    – Protawn
    Jul 17 at 18:59















Note that you can get nicer spacing around the vertical bars in conditional probabilities by using mid instead of |.
– joriki
Jul 17 at 18:56





Note that you can get nicer spacing around the vertical bars in conditional probabilities by using mid instead of |.
– joriki
Jul 17 at 18:56





1




1




oh wow! didn't know this, it indeed looks so much better this way! thank you :)
– Protawn
Jul 17 at 18:59




oh wow! didn't know this, it indeed looks so much better this way! thank you :)
– Protawn
Jul 17 at 18:59










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$$
fracpi(theta,ximid x)pi(thetamid x,xi)=fracpi(theta,xi,x)pi(x)cdotfracpi(x,xi)pi(theta,xi,x)=fracpi(x,xi)pi(x)=pi(ximid x);.
$$






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    $$
    fracpi(theta,ximid x)pi(thetamid x,xi)=fracpi(theta,xi,x)pi(x)cdotfracpi(x,xi)pi(theta,xi,x)=fracpi(x,xi)pi(x)=pi(ximid x);.
    $$






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      accepted










      $$
      fracpi(theta,ximid x)pi(thetamid x,xi)=fracpi(theta,xi,x)pi(x)cdotfracpi(x,xi)pi(theta,xi,x)=fracpi(x,xi)pi(x)=pi(ximid x);.
      $$






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        up vote
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        accepted






        $$
        fracpi(theta,ximid x)pi(thetamid x,xi)=fracpi(theta,xi,x)pi(x)cdotfracpi(x,xi)pi(theta,xi,x)=fracpi(x,xi)pi(x)=pi(ximid x);.
        $$






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        $$
        fracpi(theta,ximid x)pi(thetamid x,xi)=fracpi(theta,xi,x)pi(x)cdotfracpi(x,xi)pi(theta,xi,x)=fracpi(x,xi)pi(x)=pi(ximid x);.
        $$







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        answered Jul 17 at 18:53









        joriki

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