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)
 A: No, you cannot say A causes B.  The table you have only describes associations between A and B.  Even if you know A accurately predicted B a large percentage of the time , that does not imply that A causes B.  It may in fact, be that A causes some other, confounding variable C to occur that is highly correlated with B.
A: 
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.
A: Both of the previous answers are good, but I want to dive into the weeds on this question a little more. So we know that correlation is not causation, but correlation is also not not causation. So when do we get to say that correlation is causation. Unfortunately, the data itself can never tell us this, we can only arrive at this by imposing assumptions on the data.
Simple Example:
I am going to use directed acyclic graphs (DAGs) since they graphically encode the assumptions. Let's focus on three variables: $A$, $B$, and $U$ (you can extend this to more, but the basic concepts remain the same). $U$ is some variable we did not have the opportunity to collect. Each arrow in the DAG indicates a causal relationship, with the direction of the arrow indicating what causes what. For three variables (and the ordering restriction), following are some possible DAGs that will result in a correlation between $A$ and $B$:

Correlation is causation in only DAGs numbered 1, 2, and 3; which requires appealing to outside knowledge (although 3 is tricky since $U$ being a common cause of both $A$ and $B$ can flip the relationship from the true causal direction, e.g. $A$ is protective from $B$ in reality but $U$ makes it look harmful).
One way to determine whether correlation is consistent with causation is if we conducted a randomized experiment. If we did not randomize based on $U$ and $B$ was measured after $A$ was randomized, then we know that an arrow from $U$ to $A$ and $B$ to $A$ are implausible. Therefore, we can say that the correlation is causation. Alternatively, maybe we have some subject matter knowledge on the topic of $A$ and $B$ that says there are no common causes (unlikely in reality but this is only an example), similarly we can say that correlation is causation.
The important part is that the assumptions used to claim correlation is causation are supported by outside knowledge. How and exactly what outside knowledge is needed is an important issue.
Conclusion:
There are a variety of frameworks and formal assumptions that can be used to make the claim that a certain correlation is causation. The key part is that the data alone cannot tell you whether a correlation is or isn't causation. Some outside assumptions or procedures must be applied in order to distinguish non-causal correlations from causal correlations.
Aside:
As to my example of a scenario with causation but no correlation, DAGs are assumed to be faithful. This basically means that there are no perfect cancellations that occur (all the individual causal effects don't cancel out perfectly to result in no average causal effect). Because of this, it is a little trickier to claim that no correlation means no causation.
A: *

*Prediction means that entropy is reduced. That is, if A predicts B, then the entropy of the distribution of B is greater than the entropy of distribution B conditioned on A.


*Prediction is symmetric. If A predicts B, then B predicts A (barring degenerate cases).


*Causation is not symmetric. Causation refers to an asymmetric relationship between two events. So it follows that prediction does not mean causation.


*In the case that you present, A and B do not predict each other. While the entropy of B given A is low, it's just as low without knowing A.
