Belief networks: independence

It's not immediately obvious how to interpret the conditional relationships represented by a belief network. For example, consider the networks below.

Network(s) b, c and d represent the distribution:

\begin{aligned} p (x_{1}, x_{2}, x_{3}) & = p (x_{1} | x_{3}) p (x_{2} | x_{3}) p (x_{3}) \\ = p (x_{2} | x_{3}) p (x_{3} | x_{1}) p (x_{1}) \\ = p (x_{1} | x_{3}) p (x_{3} | x_{2}) p (x_{2}) \end{aligned}

Network(s) a represent the distribution:

p (x_{1}, x_{2}, x_{3}) = p (x_{3} | x_{1}, x_{2}) p (x_{1}) p (x_{2})

Next, consider conditional independence.

In a), $x$ and $y$ are unconditionally [...], and conditioned on $z$ they are [...]. $p (x, y | z) = p (x | z) p (y | z)$
In b), $x$ and $y$ are unconditionally [...], and conditioned on $z$ they are [...]. $p (x, y | z) \propto p (z | x) p (x) p (y | z)$
In c), $x$ and $y$ are unconditionally [...], and conditioned on $z$ they are [...]. $p (x, y | z) \propto p (z | x, y) p (x) p (y)$
Id d), $x$ and $y$ are unconditionally [...], and conditioned on $z$ or $w$ , are [...]. See book for full equation.

14.11.2019