They're not independent because there's an active path from Flu Shot to Sore Throat.
The key is that when you learn P(flu-shot) = 0, having observed sorethroat, you have to update your prior on P(exposed-to-flu) to very high,
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And that update switches the relative positions of P(sore-throat|flu-shot) and P(sore-throat|~flu-shot).
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P(S|F,X) > P(S|~F,X) P(S|F,~X) > P(S|~F,~X) P(F|S) > P(X|S) Observe S: infer F, ~X "S because F" Then observe ~F: infer X "S because ~F"
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this is one of my favorite sufjan songs
End of conversation
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