Consider the following joint distribution for the random variables A and B:
\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)
Answer
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.
Edit: To address one of the points in the comments:
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)
[1] 0
Clearly Y is caused by X, yet the correlation is zero.
Attribution
Source : Link , Question Author : elemolotiv , Answer Author : Robert Long