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I understand examples where two nodes can be dependent but conditionally independent given their common cause. For instance the common cause is high temperature, and the children nodes are high icecream sales and more number of sharks. The children are correlated through the high temperature, but independent given the high temperature. Mathematically, this is equivalent to - \begin{align*} b&=a+\epsilon_1\\ c&=a+\epsilon_2 \end{align*} $b$ is correlated with $c$, but $b\perp c|a$

I cannot imagine a scenario where cycles would occur. What is a physical or mathematical example that leads to a graph with cycles?

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Cycles in a probabilistic graphical model (PGM) violate the condition that the product of the individual node's local distributions P(v_i | parent_of(v_i) ) equal the joint of the node variables in the network P( v_1..v_n). Similarly if, one thinks causally, by analogy it looks like a case of circular reasoning.

One closely related model, a continuous time Bayes network, does allow cycles, but they "unwind" to create a temporal dependence.

In short, if you conceive of cycles in a PGM, its probably a temporal effect where each time around the cycle represents a different time-stage.

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