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This was my thought as well. I don't like showing the scatterplots in these examples, as "correlation" I think is more associated with the correlation coeffici

This was my thought as well. I don't like showing the scatterplots in these examples, as "correlation" I think is more associated with the correlation coefficient than the more generic independence that the author means in this scenario. E.g. a U shape in the scatterplot may have a zero correlation coefficient but is not conditionally independent.

bdjsiqoocwk 2y ago
> E.g. a U shape in the scatterplot may have a zero correlation coefficient but is not conditionally independent. Ok this is correct, but has nothing to do with causality. Whether or not two variables are correlated and whether or not they are independent, and when one does or doesn't imply the other, is a conversation that can be had without resorting to the concept of causality at all. And in fact that's how the subject is taught at an introductory level basically 100% of the times.
- cubefox 2y ago
  > Ok this is correct, but has nothing to do with causality. It does. Dependence and independence have a lot to do with causation, as the article explains. > Whether or not two variables are correlated and whether or not they are independent, and when one does or doesn't imply the other, is a conversation that can be had without resorting to the concept of causality at all. Yes, but this is irrelevant. It's like saying "whether or not someone is married is a conversation that can be had without resorting to the concept of a bachelor at all". You can talk about (in)dependence without talking about causation, but you can't talk in detail about causation without talking about (in)dependence.
ordu 2y ago
From the article: > NB: Correlated does not mean linearly correlated > For simplicity, I have used linear correlations in all the example R code. In real life, however, the pattern of correlation/association/mutual information we should expect depends entirely on the functional form of the causal relationships involved.
- dash2 2y ago
  The standard mathematical definition of correlation means linear correlation. If you are talking about non-independence, it would be better to use that language. This early mistake made me think the author is not really an expert.
  - cubefox 2y ago
    What is an appropriate measure of (in)dependence though, if not Pearson correlation? Such that you feed a scatter plot into the formula for this measure, and if the measure returns 0 dependence, the variables are independent.
    - currymj 2y ago
      it's a tough problem. there are various schemes for estimating mutual information from samples. if you do that and mutual information is very close to zero, then I guess you can claim the two rvs are independent. But these estimators are pretty noisy and also often computationally frustrating (the ones I'm familiar with require doing a bunch of nearest-neighbor search between all the points). I agree with the OP that it's better to say "non-independence" and avoid confusion, at the same time, I disagree that linear correlation is actually the standard definition. In many fields, especially those where nobody ever expects linear relationships, it is not and everybody uses "correlated" to mean "not independent".
      - cubefox 2y ago
        Yeah. It would be simpler to talk about causal graphs if the nodes represented only events instead of arbitrary variables, because independence between events is much simpler to determine: X and Y are independent iff P(X) * P(Y) = P(X and Y). For events there also exists a measure of dependence: The so-called odds ratio. It is not influenced by the marginal probabilities, unlike Pearson correlation (called "phi coefficient" for events) or pointwise mutual information. Of course in practice events are usually not a possible simplification.
  - 2y ago
currymj 2y ago
This is a separate issue and also a good point. Correlation sometimes means “Pearson’s correlation coefficient” and sometimes means “anything but completely independent” and it’s often unclear. In this context I mean the latter.

bdjsiqoocwk 2y ago

> E.g. a U shape in the scatterplot may have a zero correlation coefficient but is not conditionally independent. Ok this is correct, but has nothing to do with causality. Whether or not two variables are correlated and whether or not they are independent, and when one does or doesn't imply the other, is a conversation that can be had without resorting to the concept of causality at all. And in fact that's how the subject is taught at an introductory level basically 100% of the times.

cubefox 2y ago
> Ok this is correct, but has nothing to do with causality. It does. Dependence and independence have a lot to do with causation, as the article explains. > Whether or not two variables are correlated and whether or not they are independent, and when one does or doesn't imply the other, is a conversation that can be had without resorting to the concept of causality at all. Yes, but this is irrelevant. It's like saying "whether or not someone is married is a conversation that can be had without resorting to the concept of a bachelor at all". You can talk about (in)dependence without talking about causation, but you can't talk in detail about causation without talking about (in)dependence.

ordu 2y ago

From the article: > NB: Correlated does not mean linearly correlated > For simplicity, I have used linear correlations in all the example R code. In real life, however, the pattern of correlation/association/mutual information we should expect depends entirely on the functional form of the causal relationships involved.

dash2 2y ago
The standard mathematical definition of correlation means linear correlation. If you are talking about non-independence, it would be better to use that language. This early mistake made me think the author is not really an expert.
- cubefox 2y ago
  What is an appropriate measure of (in)dependence though, if not Pearson correlation? Such that you feed a scatter plot into the formula for this measure, and if the measure returns 0 dependence, the variables are independent.
  - currymj 2y ago
    it's a tough problem. there are various schemes for estimating mutual information from samples. if you do that and mutual information is very close to zero, then I guess you can claim the two rvs are independent. But these estimators are pretty noisy and also often computationally frustrating (the ones I'm familiar with require doing a bunch of nearest-neighbor search between all the points). I agree with the OP that it's better to say "non-independence" and avoid confusion, at the same time, I disagree that linear correlation is actually the standard definition. In many fields, especially those where nobody ever expects linear relationships, it is not and everybody uses "correlated" to mean "not independent".
    - cubefox 2y ago
      Yeah. It would be simpler to talk about causal graphs if the nodes represented only events instead of arbitrary variables, because independence between events is much simpler to determine: X and Y are independent iff P(X) * P(Y) = P(X and Y). For events there also exists a measure of dependence: The so-called odds ratio. It is not influenced by the marginal probabilities, unlike Pearson correlation (called "phi coefficient" for events) or pointwise mutual information. Of course in practice events are usually not a possible simplification.
- 2y ago

currymj 2y ago

This is a separate issue and also a good point. Correlation sometimes means “Pearson’s correlation coefficient” and sometimes means “anything but completely independent” and it’s often unclear. In this context I mean the latter.