1. Bayes Ball and D-Separation Examples

In the last lecture, we learned about Bayes ball rules, the D-separation and an example of how the rules is applied in the Alarm model. In order to better understand it, we will introduce a few more examples with Bayes ball rules. Keep in mind that there are more than the CIs directly asserted when we draw a directed graph, and without the help of the Bayes ball rule, it would be difficult for us to find those implied CIs.

Recall the Bayes Ball Algorithm

Question: Given a graphical model, is the statement of $\mathsf{x}_i \perp \mathsf{x}_j\space | \space \mathsf{x}_A$ true?

Note: $\mathsf{x}_A$ is a set of nodes ($A$ is a set of some variables). If $\mathsf{x}_i$ and $\mathsf{x}_j$ are conditionally independent, then they are called d-separated.

The algorithm:

1. Take the graph and color the nodes in $\mathsf{x}_A$. A colored node means that it is observed. Start a ball ("Bayes ball") bouncing from $\mathsf{x}_i$.

2. "Bounce" the ball with the six Bayes ball rules.

3. If a ball can start from $\mathsf{x}_i$ and hit/reach $\mathsf{x}_j$, then $\mathsf{x}_i$ and $\mathsf{x}_j$ are dependent, which means that the CI ($\mathsf{x}_i \perp \mathsf{x}_j\space | \space \mathsf{x}_A$) is not true.

4. If it is impossible to hit $\mathsf{x}_j$ starting from $\mathsf{x}_i$, then $\mathsf{x}_i$ $\perp \mathsf{x}_j$ | $\mathsf{x}_A$ is true in the directed model.

   Rule 1:

   

   Rule 2:

   

   Rule 3:

   

   Rule 4:

   

   Rule 5:

   

   Rule 6:

   

   Two very critical things to keep in mind are:

   • Left and right nodes in the rules which are symmetric could represent the same node.
   • Coloring of the outer nodes (node A or C) doesn't matter.

More Examples:

Example 1: Let's consider a city where we focus on the temperature, fire events and insurance claims. It's reasonable to think that the city sometimes get hot (the temperature is too high), and the chances of fire will follow to increase, so is the insurance claims. By that logic, we draw the following directed model:

                                    **Fig 7**: Temperature is **not** independent of Insurance

Question: $\mathsf{Temperature} \perp \mathsf{Insurance} ?$

With the help of Rule 1 in Fig 1, we can assert that Temperature is not independent of Insurance. Intuitively, if it is hot for a month then it is more likely to catch fire which will increase the chances of an insurance claim.

                                             **Fig 8**: $\\mathsf{Temperature} \\perp \\mathsf{Insurance} | \\mathsf{Fire}$