Is this causation?

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

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