I've heard people use the term spurious correlation in so many different instances and various ways, that I'm getting confused. Moreover, the Wikipedia page for Spurious relationship states:
“In statistics, a spurious relationship or spurious correlation is a mathematical relationship in which two or more events or variables are not causally related to each other (i.e. they are independent), yet it may be wrongly inferred that they are, due to either coincidence or the presence of a certain third, unseen factor”
A couple of observations:
Mathematically speaking, two variables $A, B$ are independent $\iff P(A | B) = P(A)$, correct?
Clearly, if two variables are correlated, even if the dependency is driven by some third factor, the two are still not independent, like the Wikipedia article claims. What's up with that?
If the “spurious” correlation is statistically significant (or not a result of coincidence), then what's wrong with that? I've seen people jumping out like rabid animals, foam coming out of their mouth screaming: “Spurious! Spurious!”.
I don't understand why they do it — no one is claiming that there is a causal link between the variables. Correlation can exist without causation, so why label it “spurious”, which is sort of equivalent to calling it “fake”?