Question

2) Let Yi,., Yn be iid N(a,a2). Let a~ known Find the posterior distribution p(u|3y). This distribution will depend N (8, T2) and trcat o2, 6, and r2 as fixed and on 2, 6, and 2. Calculations are tedious here. Use the hints given in class and follow through

Answer 1

This is the example of bayesian problem,

To calculate the p(u/y) which is posterior distribution we have formula,

p(u/y)=(p(y/u)*p(u))/p(y)

so first we need to calculate p(y/u)

here prior distribution is $\mu$~ N($\delta$,$\tau$^2)

The probability density function of Yi is,

p(Yi/$\mu$) = (2
71
$\sigma$^2)^(-1/2) exp(-(yi -$\mu$)^2/2$\sigma$^2)

since y1,y2...yn are independent,the likelihood is,

p(y/u)=$\prod_{}^{}$p(Yi/$\mu$)= (2
71
$\sigma$^2)^(-n/2) exp(-$_{}^{}\sum$(yi -$\mu$)^2/2$\sigma$^2)

The prior is,

p(u) =(2
71
$\tau$^2)^(-1/2) exp(-($\mu$-$\delta$)^2/2$\tau$^2)

p(y)=(2
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$\sigma$^2)^(-1/2) exp(-(y-$\mu$)^2/2$\sigma$^2)

p(u/y)=(p(y/u)p(u)/p(y)

   (p(y/u)p(u)= [(2
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$\sigma$^2)^(-n/2)*(2
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$\tau$^2)^(-1/2) exp[-1/2($_{}^{}\sum$(yi -$\mu$)^2/$\sigma$^2)+ exp(-($\mu$-$\delta$)^2/$\tau$^2))]]

by solving square brackets in exponential and adding and subtracting by [($\delta$/$\tau$^2)+($_{}^{}\sum$yi/$\sigma$^2)]^2 / [(1/$\tau$^2)+(n/$\sigma$^2)]

here terms involving $\delta$^2 is ((1/$\tau$^2)+(n/$\sigma$^2))$\delta$^2

and another terms involving $\delta$ is -2(($\mu$/$\tau$^2)+($_{}^{}\sum$yi/$\sigma$^2))$\delta$

remaining terms are ($\delta$^2/$\tau$^2)+($_{}^{}\sum$yi^2/$\sigma$^2)

after solving these we define some terms .,

$\tau$" =((1/$\tau$^2)+(n/$\sigma$^2)^(-1/2)) and $\delta$" =[($\delta$/$\tau$^2)+($_{}^{}\sum$yi/$\sigma$^2)]/[(1/$\tau$^2)+(n/$\sigma$^2)]

the expression which comes in bracket of exponent is ,

{(($\mu$-$\delta$")/$\tau$")^2 +[($\delta$^2/$\tau$^2)+($_{}^{}\sum$yi^2/$\sigma$^2)-(($\delta$/$\tau$^2)+($_{}^{}\sum$yi/$\sigma$^2)]^2 / [(1/$\tau$^2)+(n/$\sigma$^2)])]}

posterior pdf for $\mu$ is

p(u/y)=(2
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$\tau$"^2)^(-1/2) exp(-($\mu$-$\delta$")^2/2$\tau$"^2)

posterior distribution is normal with mean $\delta$" and variance $\tau$"^2

NOTE: Solve the exponent brackets as per given instruction above,I've given all the possible terms which will come in exponent brackets.