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Example of E-M: Genetic Linkage Problem
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Rao (1973) considers 197 animals randomly divided into four categories
as follows:
\[
                Y = (y_1, y_2, y_3, y_4)^T = (125, 18, 20, 34)^T
\]
with cell probabilities
\[
                ( 1/2 + \theta/4, (1-\theta)/4, (1-\theta)/4, \theta/4 )^T
\]
The observed-data likelihood is 
\[
        g(Y|\theta) 
        \propto \cdot (2 + \theta)^{y_1}(1-\theta)^{y_2+y_3} \theta^{y_4}
\]
If we ``augment'' the data by splitting the first category into two, 
\begin{eqnarray*}
         X & = & (x_1, x_2, x_3, x_4, x_5)^T \\
\mbox{with probabilities} & & (1/2, \theta/4, (1-\theta)/4,
                              (1-\theta)/4, \theta/4 )^T 
\end{eqnarray*}
the complete-data likelihood is now simpler:
\[
        f(X|\theta) \propto \theta^{x_2+x_5} (1-\theta)^{x_3+x_4}
\]
The ``missing data'' $Z$ is like a coin toss for which category each
observation of $y_1$ gets split into:
\[
        (Z|Y,\theta) \sim Bin(y_1, 2/(2+\theta))
\]
Now we compute the $Q$-function,
\begin{eqnarray*}
        Q(\theta,\theta^k) & = & E[(x_2+x_5)\log(\theta) + (x_3 +
        x_4)\log(1-\theta) ~|~ Y,\theta^k] \\
        & = & (y_1\theta^k/(2+\theta^k) + x_5)\log(\theta) +
        (x_3+x_4)\log(1-\theta) 
\end{eqnarray*}

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