What expresses if/how two variables are dependent on each other?

Question

The accepted answer to question Why is a correlation matrix symmetric? includes this:

The correlation matrix is a measure of linearity. It does not express how two variables are dependent on each other.

My question is : What is there that is related to if/how two variables are dependent on each other?

Edit 1 : When asking this question, not considering anything related to correlation, only want to know if there are ways to see if two or more variables have dependency on each other or not, irrespective high or no correlation between them.

I used the correlation tag because I could not think of what tag to use, please feel free to edit/correct the tag.

Answer 1

There are two senses in which one might understand the sentence that correlation does not express dependence. One is, as stated in the answer above, that correlation is symmetric, that correlation is symmetric, it doesn't have an arrow and can't be used to show that one variable has a causal effect on the other. In general, there isn't any way in which to determine causality merely from probability distributions. A good reading on the subject of causal reasoning and relation to ML is https://www.inference.vc/untitled/

The other sense might be that correlation only captures linear relations. In this ense there is a more general measure which is Mutual Information, but it requires knowledge of the probability distribution and not only a set of observations https://en.wikipedia.org/wiki/Mutual_information

Answer 2

I think what the clarification has in mind is that correlation does not imply causation, that is, that two variables are mathematically correlated, does not mean they are dependent on each other.

Here a wikipedia article if you'd like to expand on reading.

Here is a page full of examples where two variables, although highly correlated, are very clearly independent one from the other. For example: Per capita consumption of chicken vs Total US crude oil imports, or Per capita consumption of mozzarella cheese vs Civil eng. doctorates awarded.

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Having shared my understanding of the clarification, now going back to your question:

What is there that is related to if/how two variables are dependent on each other?

I think what does show if two variables are dependent on each other is not correlation alone, but a further analysis based on common sense.

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Hope this helps.