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I am studying about Bayesian Networks for the first time from a particular source. It gives the example of the following Bayesian Network enter image description here

Now I understand this network, then they go on with explaining Pearl's Network Construction algorithm. They have given examples on how the network will change if the ordering of the nodes presented to the algorithm is different. Let's take the node names to be their first letters in capitals $<P, S, C, X, D>$ . Now suppose we present the following order to this algorithm $<D, X, C, P, S>$ . So the network becomes now

enter image description here

I am not able to understand the addition of the final node. When we add $S$ to the network it says

When the final node, Smoker, is added, not only is an arc required from C to S, but another from P to S. In our story S and P are independent, but in the new network, without this final arc, P and S are made dependent by having a common cause, so that effect must be counterbalanced by an additional arc.

Can anyone help me out here. How does this get counterbalanced, maybe some form of equations to show that. Thanks !

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Firstly, in adding the $S$ node last we need an arrow from $C$ to $S$ to indicate the direct dependence between the two.

To finish there is wrong because, as stated in the text, $P,S$ having a common parent means $P,S$ are dependent, which is false. It also implies that $P,S$ are conditionally independent given $C$, and this is also false.

The only way to resolve this is to add an arrow $P$ to $S$. Note that, in any situation, an arrow from one node to another doesn't force dependence between them - they could be dependent or independent depending on the actual probability values in the conditional probability table you provide. That is, the arrow gives you the option or ability to make them dependent.

With this arrow, you will have a conditional probability table with entries such as $P(S=T\mid C=T,\; P=L)$ instead of just $P(S=T\mid C=T)$ without the arrow. If these probabilities are set appropriately, (i.e. derived from the original network) you will obtain independence of $P,S$ and also conditional dependence of $P,S$ given $C$.

I think their explanation of "counterbalance" is not wrong but could be better.

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