# Is this causation?

Consider the following joint distribution for the random variables $$AA$$ and $$BB$$:

$$\begin{array} {|r|r|}\hline & B=1 & B=2 \\ \hline A=1 & 49\% & 1\% \\ \hline A=2 & 49\% & 1\% \\ \hline \end{array} \begin{array} {|r|r|}\hline & B=1 & B=2 \\ \hline A=1 & 49\% & 1\% \\ \hline A=2 & 49\% & 1\% \\ \hline \end{array}$$

Intuitively,

• if I know A, I can predict very well B (98% accuracy!)
• but I if know B, I can’t say anything about A

Questions:

• can we say that A causes B?
• if yes, what is the mathematical way to conclude that A causes B?

thank you! (and apologies for the maybe “naive” question)

can we say that A causes B?

No, this is (presumably) a simple observational study. To infer causation it is necessary (but not necessarily sufficient) to conduct an experiment or a controlled trial.

Just because you are able to make good predictions does not say anything about causality. If I observe the number of people who carry cigarette lighters, this will predict the number of people who have a cancer diagnosis, but it doesn’t mean that carrying a lighter causes cancer.

But now I wonder: can there ever be causation without correlation?

Yes. This can happen in a number of ways. One of the easiest to demonstrate is where the causal relation is not linear. For example:

> X <- 1:20
> Y <- 21*X - X^2
> cor(X,Y)
 0


Clearly Y is caused by X, yet the correlation is zero.