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I'm trying to understand the difference between causation and correlation using conditional probabilities.

From what I understand, one may quantify causation by $P(E_1|E_2) / P(E_1)$. For example, if this ratio is $> 1$, then $E_2$ increase the likehood of $E_1$.

Now I try to come up with something similar explaining correlation and I thought of : $$P\big(\left(E_1 \cap E_2\right) \cup \left(E_1^c \cap E_2^c\right)\big)$$ where $E_1^c$ is the complement of $E_1$. But I can't think of correlation in terms of conditional probability. Does correlation alone give information about conditional probability ?
Thanks for helping.

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  • $\begingroup$ Nonzero correlation between two random variables $X,Y$ just means that $E(XY) \neq E(X)E(Y)$. If the two variables are positively correlated, I think the ratio you defined would be $>1$ so I'm not sure how that captures causation but not correlation. $\endgroup$ – Macro Jul 5 '12 at 3:04
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Conditional Probability and Causality

The idea that you can define causation in terms of conditional probability was the 'probabilistic causality' programme in philosophy associated with e.g. Cartwright and Eels. Arguably, it failed (See Pearl for the argument). A good introductory read on the topic is here. Several counterexamples to the probability raising relationship you suggest are in section 2.10.

As a result, it is unlikely that you will fully understand or otherwise reconstruct the difference between correlation and causation using only the machinery of conditional probability because it is insufficient. Explicitly causal i.e. non-probability assumptions are needed in addition.

Correlation

Correlation is, as @Michael Chernick and other commenters point out, closely related to conditional probability. In a narrow technical sense it is a standardised undirected measure of linear or at least monotonic association between two variables. In a wider informal sense it is as Michael describes it: a departure from statistical independence. In either sense it may appear as a result of an underlying causal relationship. Or not, e.g. it instantiates one of the counterexamples above or exhibits Simpson's Paradox. Hence the difficulty/impossibility of reconstructing the one with the other.

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  • $\begingroup$ Thanks, the link is a good presentation on the subject, just what I was looking for. $\endgroup$ – ehmicky Jul 5 '12 at 11:47
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Showing that $E_2$ increases the likelihood of $E_1$ is just another way of saying that there is a relationship and hence a nonzero correlation between the events. It does not show causation. There are no probability relationships that "show" causation. Correlation means $P(E_1 E_2)$ not equal $P(E_1) P(E_2)$ which can be written in terms of conditional probabilities as $P(E_1|E_2) \neq P(E_1)$.

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  • $\begingroup$ Ok, so as far I get it, correlation is simple and everyone agrees upon it, whereas causation is far more debated. $\endgroup$ – ehmicky Jul 5 '12 at 11:49

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