In the bayesian network given in the figure below, what is true if A is observed?

Question

In the bayesian network given in the figure below, what is true if A is observed?

BlackTom AI · Accepted Answer

To analyze the question, we first note the network structure: A points to B and A points to C (A -> B and A -> C). Now, consider what happens when A is observed.

Option 1: Flow of information goes from A to B and C. This is true because A is a parent of both B and C, so observing A provides information about both downstream nodes B and C. In probabilistic terms, P(B|A) and P(C|A) are influenced by the value of A, so information flows from A to each of its children.

Option 2: C is conditionally independent of B. When we condition on A, B and C become independent given A in this structure: B ⟂ C | A. This reflects the common-parent phenomenon in Bayesian networks: once A is known, the only shared influence between B and C is via A, which has been conditioned on, breaking the dependence between B and C.

Option 3: All of the possibilities are correct. Since the first two statements hold under A being observed, this composite option is consistent with the network's behavior in this scenario.

Option 4: None of the possibilities are correct. This contradicts the analysis above, as we can see at least the first two statements being true when A is observed, so this option is false.

Option 5: There is no flow of information from B to C. In the given graph, there is no direct B → C edge and no indirect path that remains active once A is observed. The key is that with A observed, the path B ← A → C is blocked for information flow from B to C, so effectively there is no information flowing from B to C when A is known. This makes the statement true in the observed-A scenario.

Overall, under A being observed, the intuitive and formal reasoning aligns with all the statements that assert information flow from A to B/C, conditional independence of B and C given A, and no active B-to-C flow, supporting the conclusion that all of the listed implications can be considered correct in this context.

In the bayesian network given in the figure below, what is true if A is observed? Single choice

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