# Why are these variables not conditionally dependent given 'active triplets' and the 'explaining away' effect?

I'm following the Udacity Intro to AI course.

This quiz gives the following Bayes network and asks whether different variables are conditionally independent or not. (The explanation of the nodes, sunny, raise etc are mine, but are similar to examples used in the course.)

On the forum someone wondered whether: $A \perp C \ | \ B$ was false.

That is: A and C are conditionally dependent given B.

Intuitively I can see how this is not the case. Given that you already know it is hot, knowledge about whether a person got a raise or not, does not give you any new knowledge about whether it is sunny.

However we were given the following diagram of active (contain dependent variables) and inactive triplets (independent variables) where shaded variables are known:

Using this diagram, in order to show that A & C are conditionally dependent given B, the argument could go as follows:

1. By active triplet #2 you can see that B & D are dependent.
2. Now that we have inferred knowledge of D, by the 'explaining away' effect, or active triplet #3, we can say that A & C are conditionally dependant.

So initially my theory was that:

1. Your knowledge of D was gleaned from A via B (active triplet #2)
2. In the second step you cannot use the 'explaining away' effect to go back up that same path to infer new knowledge of A.

However I then added a new node F as such:

Now I tried to show that A & F are conditionally dependent:

1. By active triplet #2 you can see that B & D are dependent.
2. Now that we have inferred knowledge of D, by the 'explaining away' effect, or active triplet #3, we can say that A & F are conditionally dependent.

But of course this is incorrect. Given that you know whether it is hot, knowledge that a person got a raise tells you nothing about whether they met an old friend or not.

So finally the answer I have come up with is that knowledge of a node in an 'explaining away' effect active triplet (#3), must be obtained against the causal flow... (In this case that would mean direct knowledge of D, or indirect via E.)

Or in other words if we can just 'explain away' the node using the same path via which we obtained knowledge of the node, well we get no new information.

I'm probably making a mess of what is a very simple problem, but:

1. How do I show simply that A & C are not conditionally dependent given B?
2. What exact part(s) of the arguments I provided are wrong?
3. Finally, is my understanding of the explaining away effect triplet correct?

The shaded nodes in your list of active and inactive triplets are labelled "known". I don't know whether "known" is a good choice of words or not, but what this is supposed to mean is that they have to be one of the variables we are conditioning on i.e. they have to be one of the variables to the right of the "$|$" symbol. In your case "$A\bot C|B$" the only "known" variable is $B$. So even though learning $B$ tells us something about $D$, this doesn't mean that $D$ is "known" in the sense that it would have to be in order to use active triple #3.
• Yes. The way I'd put it is that the "explaining away" effect only produces a dependence between causes of $D$ when your knowledge about $D$ has arisen by learning about $D$ directly or by learning about one of it's causes (i.e. its descendants in the graph). This is precisely what #3 and #4 are telling you. – Oscar Cunningham Nov 6 '14 at 19:21