Conditional probability and causality

Question

I would like to understand the link between conditional probabilities and causality. More precisely:

Assume we have two variables $A=\{0,1\}$ and $B=\{0,1\}$ and we observe:

$P(A=1|B=1)>P(A=1|B=0)$

i.e. the probability to observe $A=1$ increases when $B=1$ compared to the case where $B=0$.

I suspect we cannot say there is a causal effect of $B$ on $A$ just from equation (1), but I would like to properly understand why. Importantly, I would like to understand if we still can be concerned by reverse causality in this case (i.e. $A$ causes $B$ and not the opposite).

Answer 1

One reason why causality cannot be expressed by conditional probability (without further assumptions) is that you can turn $P(A|B)$ into $P(B|A)$ with Bayes theorem and causality does not go both ways (similar to the saying „correlation does not imply causation“).

An example in which neither $P(A|B)$ nor $P(B|A)$ give any causal information would be be that an event $C$ is the only possible causal event for $A$ and $B$ implying $P(A=1|C=1)= 1$, $P(A=1|C=0) = 0$, $P(B=1|C=1) = 1$, $P(B=1|C=0) = 0$. Therefore $P(A=1|B=1)=1$, $P(A=1|B=0)=0$ but by assumption $B$ has no causal influence on $A$, just a common cause.

Consider another example and this time more concrete. let‘s say $A$ is an event that you have a deadly disease and $B$ means dying this year. In addition to that there is also a chance that you die this year without having this disease. However in this imaginary world there is a super drug and all the people that have the deadly disease get it (and no one else) and it ensures that you survive this year. Therefore $P(B=0|A=1) = 1 > P(B = 0|A=0)$ even though the deadly disease certainly has causal influence on you dying.

Answer 2

I agree with Alexis that this is a complicated question best explained by textbooks in causal inference like the ones mentioned. To briefly answer your question, the reason we cannot say that there is a causal effect of $B$ on $A$ is that there are other possible explanations that are consistent with the observed data. Getting the causal ordering of the variables wrong (i.e., "reverse causality") is one way that you could observe the found association, but there are several others, including confounding (i.e., $A$ and $B$ have a common cause) and selection (i.e., you inadvertently selected on something that $A$ and $B$ both cause). These cases are covered in the books mentioned in the comments.

Answer 3

Good answers and relevant reference are already given. However it seem me that there is at least another useful point of view for clarify the problem. The idea to infer causal effects from conditioning is not new, It come from a lot years ago. Infact in some old references (maybe not only) the Bayes Theorem, the most relevant conditioning rule, was used also for try to discover the more probable cause of some effect.

Unfortunately this is not possible, Judea Pearl wrote a lot about that. We can read among others: Bayesianism and Causality, or, why I am only half-bayesian - (2001); but also the popular book: The Book of Why spent several page about the failed attempt to infer causality from conditioning.

The core of the problem is that, even if the conditioning sound like "what if" and, then, causal, the matter of fact is that conditional probability/distribution, in its proper nature, deal with passive observation while in order to achieve causal effects we need of interventions. Basically the problems explained in the other answers disappear if well managed interventions come in place. This discussion can give an idea: conditional and interventional expectation

Warning: the idea to infer causal effect from conditioning is not dead. Infact the experimental paradigm frequently used in econometrics as ideal benchmark (as if rule) make a wide use of conditioning as main tool; without ad hoc other mathematical tools. The trick is that if we add some causal assumptions to the story, the conditioning can be enough in order to deal with causality. However Pearl insisted al lot on the necessity of new and ad hoc language in order to "free the field" from ambiguity and strong limitations. I think that the future will prove him right.