In Bishop's PRML, chapter 8 is dedicated to graphical models. In figure 8.32, we have the following figure showing a directed and an undirected graph:
These two graphs are said to be "equivalent". What is meant by this? As far as I understand, the directed graph describes conditional probabilities, when an arrow from $x_1$ to $x_2$ means that $x_2$ is dependent on x1, or $p(x_2|x_1) \neq p(x_2)$ in general but $p(x_1|x_2) =p(x_1)$. And in the directed graph, the conditional probabilities can go in any direction, so in general, $p(x_2|x_1) \neq p(x_2)$ and $p(x_1|x_2) \neq p(x_1)$.
If this is the case, the two graphs can not be equivalent since graph b) clearly doesn't express the fact that $x_1$ is independent of $x_2$. But why then are they said to be equivalent? And what is the point of converting a directed graph to an undirected graph if they do not express the same thing?
Best Answer
They aren't equivalent. The caption for the second one should really say something like "the corresponding undirected graph" or "the underlying undirected graph".
The second picture as what you get if you forget the information of which way the edges are going. But this isn't an equivalence in any real sense, since you can't go backwards: if you start with an undirected graph there are many different ways to make it directed, which express different things.
The only time there is a point in converting a directed graph to an undirected graph is if the directions don't matter for a particular application.