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I was wondering if two variables can be independent and conditionally independent. For example, A and D are independent. But are they also independent given the evidence C? I think they are, because it doesn't matter what C is, knowing C won't alter probabilty of D or A.

I feel I'm missing something.

belief network

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Yes, of course variable can be conditionally and unconditionally independent (the simple example: when three variables A, B, C are independent then A is of course independent of B, but also A is independent of B given C)

In example in your graph. But one correction: knowing (observing) C indeed says nothing about D, but influences probability distribution of A as these two are dependent.

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