Exercise 14.7 [handedness-exercise]
Let $H_x$ be a random variable denoting the handedness of an individual $x$, with possible values $l$ or $r$. A common hypothesis is that left- or right-handedness is inherited by a simple mechanism; that is, perhaps there is a gene $G_x$, also with values $l$ or $r$, and perhaps actual handedness turns out mostly the same (with some probability $s$) as the gene an individual possesses. Furthermore, perhaps the gene itself is equally likely to be inherited from either of an individual’s parents, with a small nonzero probability $m$ of a random mutation flipping the handedness.
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Which of the three networks in Figure handedness-figure claim that $ {\textbf{P}}(G_{father},G_{mother},G_{child}) = {\textbf{P}}(G_{father}){\textbf{P}}(G_{mother}){\textbf{P}}(G_{child})$?
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Which of the three networks make independence claims that are consistent with the hypothesis about the inheritance of handedness?
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Which of the three networks is the best description of the hypothesis?
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Write down the CPT for the $G_{child}$ node in network (a), in terms of $s$ and $m$.
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Suppose that $P(G_{father}l)=P(G_{mother}l)=q$. In network (a), derive an expression for $P(G_{child}l)$ in terms of $m$ and $q$ only, by conditioning on its parent nodes.
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Under conditions of genetic equilibrium, we expect the distribution of genes to be the same across generations. Use this to calculate the value of $q$, and, given what you know about handedness in humans, explain why the hypothesis described at the beginning of this question must be wrong.