The main weakness of Markov networks is their inability to represent induced and non-transitive dependencies; two independent variables will be directly connected by an edge, merely because some other variable depends on both. As a result, many useful independencies go unrepresented in the network. To overcome this deficiency, Bayesian networks use the richer language of directed graphs, where the directions of the arrows permit us to distinguish genuine dependencies from spurious dependencies induced by hypothetical observations. Reiterating the example of Section 3.1.3, if the sound of a bell is functionally determined by the outcomes of two coins, we will use the network coin 1bellcoin 2, without connecting ...

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