**arc reversal**operation of in a Bayesian network allows us to change the direction of an arc $X\rightarrow Y$ while preserving the joint probability distribution that the network represents Shachter:1986. Arc reversal may require introducing new arcs: all the parents of $X$ also become parents of $Y$, and all parents of $Y$ also become parents of $X$.

1. Assume that $X$ and $Y$ start with $m$ and $n$ parents, respectively, and that all variables have $k$ values. By calculating the change in size for the CPTs of $X$ and $Y$, show that the total number of parameters in the network cannot decrease during arc reversal. (

*Hint*: the parents of $X$ and $Y$ need not be disjoint.)

2. Under what circumstances can the total number remain constant?

3. Let the parents of $X$ be $\textbf{U} \cup \textbf{V}$ and the parents of $Y$ be $\textbf{V} \cup \textbf{W}$, where $\textbf{U}$ and $\textbf{W}$ are disjoint. The formulas for the new CPTs after arc reversal are as follows: $$\begin{aligned} {\textbf{P}}(Y | \textbf{U},\textbf{V},\textbf{W}) &=& \sum_x {\textbf{P}}(Y | \textbf{V},\textbf{W}, x) {\textbf{P}}(x | \textbf{U}, \textbf{V}) \\ {\textbf{P}}(X | \textbf{U},\textbf{V},\textbf{W}, Y) &=& {\textbf{P}}(Y | X, \textbf{V}, \textbf{W}) {\textbf{P}}(X | \textbf{U}, \textbf{V}) / {\textbf{P}}(Y | \textbf{U},\textbf{V},\textbf{W})\ .\end{aligned}$$ Prove that the new network expresses the same joint distribution over all variables as the original network.

The **arc reversal** operation of in a Bayesian network allows us to change the direction
of an arc $X\rightarrow Y$ while preserving the joint probability
distribution that the network represents Shachter:1986. Arc reversal
may require introducing new arcs: all the parents of $X$ also become
parents of $Y$, and all parents of $Y$ also become parents of $X$.

1. Assume that $X$ and $Y$ start with $m$ and $n$ parents,
respectively, and that all variables have $k$ values. By calculating
the change in size for the CPTs of $X$ and $Y$, show that the total
number of parameters in the network cannot decrease during
arc reversal. (*Hint*: the parents of $X$ and $Y$ need
not be disjoint.)

2. Under what circumstances can the total number remain constant?

3. Let the parents of $X$ be $\textbf{U} \cup \textbf{V}$ and the parents of $Y$ be
$\textbf{V} \cup \textbf{W}$, where $\textbf{U}$ and $\textbf{W}$ are disjoint. The formulas for the
new CPTs after arc reversal are as follows: $$\begin{aligned}
{\textbf{P}}(Y | \textbf{U},\textbf{V},\textbf{W}) &=& \sum_x {\textbf{P}}(Y | \textbf{V},\textbf{W}, x) {\textbf{P}}(x | \textbf{U}, \textbf{V}) \\
{\textbf{P}}(X | \textbf{U},\textbf{V},\textbf{W}, Y) &=& {\textbf{P}}(Y | X, \textbf{V}, \textbf{W}) {\textbf{P}}(X | \textbf{U}, \textbf{V}) / {\textbf{P}}(Y | \textbf{U},\textbf{V},\textbf{W})\ .\end{aligned}$$
Prove that the new network expresses the same joint distribution
over all variables as the original network.