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?
 A: Let's first tackle dependence versus correlation and then we'll add causation into the mix. There are exceptions to almost every rule, some of them worth only a minimal footnote and you should start worrying about those exceptions only after you've been exposed to what is true as a rule.
Correlation is one type of dependence. Dependence is a broader concept. Correlation is a measure of linear dependence only. Two random variables are independent if they are not correlated and not dependent in any other way. Intuitively, two random variables are independent if knowledge that one has taken a particular value tells us absolutely nothing we didn't already know about the probability distribution of the other variable. Therefore, two random variables that are correlated are, by definition, dependent. With correlated variables, knowledge that one random variable has taken a particular value does tell us something new about the probability distribution of the other/correlated variable.
In practice, two causally related variables will be dependent as a rule, and a large fraction of those times they will be correlated. Causality is manifested as structural dependence. If you have drawn the correct DAG for the model and you are conditioning on the correct set of variables, if your data is without material errors, if you happen to have the correct functional form for the dependence then you will find correlation between the cause and effect variables. That is a lot of if's. Nevertheless, in practice two causally related variables will tend to be dependent, and a large fraction of those times they will be correlated.
