My Math Forum Prior And Posterior

 February 10th, 2011, 03:27 PM #1 Senior Member   Joined: Sep 2009 Posts: 115 Thanks: 0 Prior And Posterior So I am told that an unknown quantity Y has a Galenshore(a, $\theta$) distribution if its density is given by: $p(y)=\frac{2}{\Gamma(a)}\theta^{2a}y^{2a-1}e^{-\theta^2y^2}$ for $y>0, \theta>0, a>0.$ Now assume that a is known. So for this density, $E[Y]=\frac{\Gamma(a+1/2)}{\theta\Gamma(a)}, E[Y^2]=\frac{a}{\theta^2}.$ 1.First of all I am asked to identify a class of conjugate prior densities for $\theta$ and then plot a few of these. 2.Now suppose $Y_1,Y_2,...,Y_n\sim$Galenshore(a,$\theta$) and that they are i.i.d. I am to find the posterior distribution of $\theta$ given $Y_1,Y_2,...,Y_n$ using a prior from my conjugate class. 3. Then I need to determine $E[\theta|y_1,...,y_n]$ My work so far: 1. I found a Jeffrey prior by taking the log of p(y)=$log(\frac{2}{\Gamma(a)})+2alog(\theta)+(2a-1)log(y)-\theta^2y^2$ Then differentiating twice with respect to $\theta$ I get: $\frac{-2a}{\theta^2}-2y^2$ So then $-E[\frac{-2a}{\theta^2}-2y^2]=\frac{2a}{\theta^2}+\frac{2a}{\theta^2}=\frac{4a} {\theta^2}$ Then taking the square root of this I get $\frac{2\sqrt{a}}{\theta}$ So our prior is proportional to this, but this does not remind me of any distributions I know. Any help would be appreciated. To do number 2 and 3 I realize that I need to know the answer to number 1 beforehand.
 February 26th, 2011, 02:35 PM #2 Newbie   Joined: Feb 2011 Posts: 2 Thanks: 8 Re: Prior And Posterior This is the first of three so called ultimate collections of Mathematics (i have posted another ultimate collection for physics). The books included are compiled from the numerous fragmented torrents available on the net, each of which i have meticulously downloaded, fixed titles, organized by subject content and removed redundancies. I will be seeding this for the next three months everyday between 2am and 5am CST so be patient if you dont instantly record a substantial download speed. ================== Law Ebooks
 November 11th, 2011, 03:32 AM #3 Newbie   Joined: Nov 2011 Posts: 1 Thanks: 0 Re: Prior And Posterior I am with the same problem, someone can help me?
 February 27th, 2013, 08:37 AM #4 Newbie   Joined: Feb 2013 Posts: 1 Thanks: 0 Re: Prior And Posterior did you ever get an answer to this?
February 27th, 2013, 05:36 PM   #5
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Re: Prior And Posterior

Quote:
 Originally Posted by dl338 did you ever get an answer to this?

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