That doesn't, go through. The subset, because otherwise the subset

would separate it. So, it doesn't go through SIJ.

So there needs to be an edge that goes for example from here to there or it

doesn't matter which node I pick, which pair of nodes I pick.

There needs to be some node on the green side and some node on the blue side.

This is the green side and this is the blue side and an edge that goes between

them that doesn't go through the subset. But now that implies that there needs to

be some factor. That involves those two variables.

So in this case, e comma h. But now because of family preservation

that factor must sit in some. One of the clique's in this clique tree.

And that clique is either on the green side or the blue side.

It has to be somewhere. So, lets assume that we put that clique

without loss of generality on. The green site that's not what happens we

have an H that's sitting here and remember H is a blue variable and H is

also sitting here because it is on the blue side and now we have an H in one

clique and an H in the other and running intersection property tells us that H

needs to be everywhere in between and specifically.

It needs to be in the subset, which is a violation of the assumption, either of

the running interception property or of the assumption that the, the variable h

is not in this in the subset. And so that's, a sort of, a somewhat

formal proof outline that can, with a little bit of extra effort and notation,

be made into a rigorous proof of why a running intersection property implies

this independence assumption. And I didn't prove the other direction,

because this is actually the direction that we care about more.

So we start out this whole discussion by saying that these properties have

computational implications. But where are these computational

implications? What can we conclude from the fact that

the subset needs to separate the graphics to conditionally independent pieces?

Well it turns out that in many graphs that implies a certain minimal complexity

that can sometimes be quite large. So let's do this in, let's look at this

in the context of two quite simple but very commonly used examples.

So here is an example of A. What's called a, a complete bipartite

gra, graph. Where we have two sets of variables.

That have no edges between each of the set separately.

But where all of the cross edges are present.

This is a structure that is that we've actually seen before.

We saw it in the context of the, play model for a student for course

difficulty. And student intelligence where we had a

bunch of difficulty variables, a bunch of intelligence variables and there were no

edges between the difficulties or, I mean between the intelligences.

But for every difficulty intelligence pair there was an edge that was induced

by the V structure corresponding to an observed student grade between the course

difficulty, corresponding course difficulty and not student intelligence.

Now what is the smallest subset that we could possibly construct for this graph?

Can we, for example just look at, say these two As and separate out the graph

into two conditionally independent pieces?

Well, no, not really because for example if we now look at these two Bs we see

that you can connect them via any one of the other As that I didn't include in the

subset, and so there is no, this doesn't de-couple the graph at all.

With a little bit of extra thought it's not difficult to convince oneself that

the smaller sets that we could construct, that actually breaks up the graph into

meaningful pieces is either all the variable son the one side or all the

variables on the other. Which means the smaller sub set.