92 CHAPTER 3. BAYESIAN NETWORKS 3.6 Entailed Conditional Independencies* If (G, P) satisfies the Markov condition, then each node in G is conditionally independent of the set of all its nondescendents given its parents. Do these con- ditional independencies entail any other conditional independencies? T h a t is, if (G, P ) satisfies the Markov condition, are there any other conditional indepen- dencies which P must satisfy other than the one based on a node's parents? The answer is yes. Such conditional independencies are called entailed conditional independencies. Specifically, we say a DAG e n t a i l s a conditional indepen- dency if every probability distribution, which satisfies the Markov condition with the DAG, must have the conditional independency. Before explicitly show- ing all entailed conditional independencies, we illustrate that one would expect them. 3.6.1 Examples of Entailed Conditional Independencies Suppose some distribution P satisfies the Markov condition with the DAG in Figure 3.24 (a). Then we know Ip(C, {F, G}I , B) because B is the parent of C, and F and G are nondescendents of C. Furthermore, we know Ip(B, GIF ) because F is the parent of B, and G is a nondescendent of B. These are the only conditional independencies according to the statement of the Markov condition. However, can any other conditional independencies be deduced from them? For example, can we conclude Ip(C, GIF)? Let's first give the variables meaning and the DAG a causal interpretation to see if we would expect this conditional independency. Suppose we are investigating how professors obtain citations, and the vari- ables represent the following: G: F: B: C: Graduate Program Quality First Job Quality Number of Publications Number of Citations. Further suppose the DAG in Figure 3.24 (a) represents the causal relationships among these variables, and there are no hidden common causes. 6 Then it is reasonable to make the causal Markov assumption, and we would feel that the probability distribution of the variables satisfies the Markov condition with the DAG. Suppose we learn that Professor La Budde attended a graduate program (G) of high quality. We would now expect his first job (F) may well be of high quality, which means that he should have a large number of publications (B), which in turn implies he should have a large number of citations (C). Therefore, we would not expect Ip(C, G). Suppose we next learn that Professor Pellegrini's first job (F) was of high quality. T h a t is, we instantiate F to "high quality." The cross through the node F in Figure 3.24 (b) indicates it is instantiated. We would now expect his 6We make no claim that this model accurately represents the causal relationships among the variables. See [Spirtes et al., 1993, 2000] for a detailed discussion of this problem.