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Let's consider the simple network below:simpleNetwork1.png Isn't the same conditional independence relation captured by either of the serial connections where there is[*:16w6m7ny]a directed edge from C to A and a directed edge from A to B, or [/*:m:16w6m7ny][*:16w6m7ny]where there is a directed edge from B to A and a directed edge from A to C?[/*:m:16w6m7ny]Aren't these three connections score equivalent, and if so can (the different) learning algorithms distinguish between them?
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This network represents the following probabilistic dependencies and independencies:[*:1piz2ttg]A and B are directly dependent,[/*:m:1piz2ttg][*:1piz2ttg]A and C are directly dependent,[/*:m:1piz2ttg][*:1piz2ttg]B and D are directly dependent,[/*:m:1piz2ttg][*:1piz2ttg]A and D are dependent through B and then conditionally independent given B,[/*:m:1piz2ttg][*:1piz2ttg]B and C are dependent through A and are then conditionally independent given A,[/*:m:1piz2ttg][*:1piz2ttg]C and D are dependent through A and B and then conditionally independent given A or B.[/*:m:1piz2ttg]The exact same dependencies and independencies can be represented with these 3 equivalent networks:equivalentNetwork.png These three networks plus the original one belong to the same equivalence class, represented with a network that does not have any oriented arc.networkOrientedArc.png You can get this graph in BayesiaLab with Validation mode: Analysis | Graphic | Show the EdgesThe BayesiaLab Unsupervised Structural Learning algorithms are based on the Minimum Description Length (MDL) score. This score will be identical for these four networks. Without any external (expert or temporal) information, BayesiaLab will then do not have any way to rationally choose among these networks.Note that these equivalences are only true in the framework of observational inference. The direction of the arcs will indeed matter if we consider these networks as Causal Bayesian networks. In this case, none of these networks will be equivalent.
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