Bayesian Networks: Conditional Independence
++ During the lectures on Bayesian networks, you were already introduced to the concept of conditional (in)dependence. + To visualize this, we can use the program Bayes Server. + Follow this link to navigate to the online application. + If you prefer an offline version, you can download it here. +
+ When you start up the Bayes Server, you see all kinds of predefined networks. + Note that you can also create your own network. For now, however, it suffices to open an existing network. + Now we will first illustrate some examples of conditional independence with the Asia network. + +
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+ As you can see, the network has eight different variables, each with their own probabilities as shown in the bar graphs. + At the top, you can click on analyze and then D-Separation → D-Separation Display. +
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+ At that point a window appears where you can select the variable that will serve as the source node. + The D-separation algorithm will try to find a path from this node to all the others. + Is there an unblocked path to a node? If so, it colors green. Otherwise, it colors red. + A green node and a red node are respectively conditionally dependent and conditionally independent of the selected source node. + Now we can play around a bit by clicking on the bar graphs in the nodes. + When clicked on, these nodes become evidence. + This allows us to see which nodes become / no longer are reachable when an intermediate node becomes evidence. +
+ +Now think back to the lectures. There are a total of four scenarios in which a path is blocked:
+When the intermediate node is no evidence and the pattern forms a collider (both A and B point to C).
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+ When the intermediate node is evidence and the pattern forms a fork (C points to A and B).
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+ When the intermediate node is evidence and the pattern forms a chain (A points to C and C points to B).
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+ When the intermediate node is evidence and the pattern forms a chain (B points to C and C points to A).
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+ Practice for yourself:
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- + Try reproducing the four scenarios in the opened Asia network. + Before you click the "D-Separation Display" button, try to imagine which nodes will turn green and which will turn red. + Do your expectations match the results? + +
- + How does the result change if you select multiple nodes as source nodes? + +
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+ A possible example of a collider in the Asia network is the following: + Choose Has Lung Cancer as source node. + Remember that in the case of a blocked path with a collider, the intermediate node should not be a evidence. + With Tuberculosis Or Cancer as an intermediate node, we see that the path to Has Tuberculosis is blocked. +
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+ A possible example of a fork in the Asia network is the following: + Choose Has Lung Cancer as source node. + Then choose Smoker as evidence by clicking on the bar graph. + Next, you see that the previously accessible node Has Bronchitis is now no longer accessible. +
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+ A possible example of a chain in the Asia network is the following: + Choose Visit to Asia as source node. + Then choose Has Tuberculosis as evidence by clicking on the bar graph. + Next, you see that the previously accessible node Tuberculosis or Cancer is now no longer accessible. +
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+ Finally, we look at what happens when we select multiple nodes as source nodes: + We choose Visit to Asia and Smoker as source nodes. + Then pick Has Tuberculosis as evidence by clicking on the bar graph. + Now take a look at Tuberculosis or Cancer. + There is a blocked path from Visit to Asia to Tuberculosis or Cancer but because the path from Smoker to Tuberculosis or Cancer is not blocked, this node is still conditionally dependent. + So if there is an unblocked path from one of the source nodes to a node, then that node is considered conditionally dependent on the source nodes. +
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+ Don't hesitate to experiment with other networks yourself. + Try to find examples of the four scenarios in other networks. + This way, your understanding of conditional independence can only increase. +
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