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\title{Stat 471: Homework 05}
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\author{Moo K. Chung\\
mchung@stat.wisc.edu}
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{\bf Due Date:} November 24, 1:20pm. No late assignments will be
accepted or graded. There will be $10\%$ bonus points for
exceptional solutions.


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\begin{enumerate}

\item Simulate bivariate normal
$${\bf Z} =(X,Y)' \sim N(0, \Big[\begin{array}{cc}
                                     \sigma^2 & \rho\\
                                     \rho & \sigma^2 \end{array}\Big])$$
when $(\rho,\sigma)=(0.9, 0.5)$ using the Gibbs sampler. Plot your
Markov chains.

 \item $(3,2,4,2,1)$ are observations for model
 $$Y_i = \beta + \epsilon_i.$$ $\epsilon_i \sim N(0,1)$ and prior $\beta|\gamma \sim N(\gamma,1)$, $\gamma
 \sim N(2,1)$. Estimate $\beta$ based on the Gibbs sampling
 method.


 \item From $\tt{strengh.data}$, fit the model
$${\tt grip}_i = \beta_0 + \beta_1 {\tt arm}_i + \epsilon_i,$$
using the Gibbs sampler. $\epsilon_i \sim N(0,3^2)$ and the priors
are $\beta_0 \sim N(110, 3^2)$ and $\beta_1 \sim N(1,1)$. Compare
your result with the least squares estimation method we studied.

\item Download data
$\tt{http://www.stat.wisc.edu/~mchung/teaching/data/bainslice2}$
and run the following codes.
\begin{verbatim}
load brainslice2; intensity=reshape(brainslice2,256*124,1);
intensity=intensity(find((intensity >=50)&(intensity<=250)));
imagesc(brainslice2);colormap('bone');colormap;
hist(intensity,100);
\end{verbatim}
You should get 19,660 values for $\tt{intensity}$ which are gray
scale values between 50 and 250 of the magnetic resonance image
(MRI) of the human brain. Let $y_1,\cdots ,y_{19660}$ be the
observations. From the histogram below, we see that the image
intensity can be modelled as the mixture of normals such that the
probability density is given by
$$f(y) = p f_1(y) + (1-p)f_2(y) \mbox{   : two component Gaussian mixture model}$$
where $f_1 \sim N(\mu_1,\sigma_1^2)$ and $f_2 \sim
N(\mu_2,\sigma_2^2)$. Assume $\mu_1=170,\sigma^2=20^2, \mu_2=210,
\sigma_2^2=10^2$. Estimate parameter $0<p<1$ by maximizing the
likelihood function using the EM algorithm. Use both deterministic
and Monte-Carlo versions.  What this parameter $p$ measure?
\end{enumerate}
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