# Correlation vs dependence vs causality

What are the relationships between correlation, dependence and causality?

I know that non-zero correlation does not imply causality. Though obviously causality implies correlation.

I also know that independence implies zero correlation.
By logical contraposition, if $$P \rightarrow Q$$, then $$\neg Q \rightarrow \neg P$$.
Does it mean then that non-zero correlation implies dependence?

What is the relationship between dependence and causality?
Does causality imply dependence?

• "Obviously causality implies correlation:" that isn't so obvious. In fact, feedback loops are designed to prevent just that! A technical nicety--but an important one--is that correlation doesn't necessarily exist, so in that sense independence doesn't always imply zero correlation, either. But that doesn't ruin your argument: nonzero (but finite, defined) correlation and independence cannot both occur.
– whuber
Feb 11 at 22:55
• Causality implies mutual information. Might be a correlation, might not. Feb 12 at 3:43
• @AdrianKeister, what if there are compensating effects such that two causes of equal strength and opposite effect yield the outcome unchanged? Feb 12 at 6:50
• Regarding though obviously causality implies correlation (which is incorrect), let me quote Mark Twain: It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so. One of my professors used to say: if you want to find a mistake in a proof, look at the place where it says trivially or obviously or clearly :) Feb 12 at 6:53
• Yes, I understand now that causality does not imply correlation. For example if we have pseudo-random generator, which takes a number and spits out another evenly distributed number, then we have no correlation, but we know that pseudo-random generator "caused" the random number. Another less extreme example would be a sinusoidally related physical variables, where independent value is directly related to the dependent value, but the correlation of the points on the sine curve is zero. My main question though is "does non-zero correlation imply dependence?". Feb 12 at 7:19