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So I have a bunch of parameters say $X_i \in \mathbb{X}$, where $i\in \{0,1,2...N\}$ . Now, I can find the correlation between any two features $i,j$ and get a matrix $\mathbb{C} = \text{Corr}(X,X)$.

But if I need to find the causation say between any feature $i,j$ i.e. if $i$ occur due to $j$ or not etc.

  1. How can I do that mathematically?

  2. Also, how to represent it mathematically (much like correlation values of 0 to 1)?

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    $\begingroup$ Well, first we need an accepted quantification of causation. There is none but for the sake of argument let's say it is Transfer Entropy. Then we need to normalise it across the variables in our sample. Again, a bit fuzzy but can be done. We have to also smooth out some little niggles like the fact our matrix will be non-symmetric. And then we have a "causation matrix". Read on Information & Transfer Entropy first, then revisit this. $\endgroup$
    – usεr11852
    Commented Mar 21, 2022 at 18:15
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    $\begingroup$ @AdrianKeister, I wonder what $P(X_i)$ stands for and what properties it has (keeping in mind it is an element of a matrix "much like a correlation matrix"). user11852, I am not familiar with transfer entropy from before, but it does not seem it is a causal concept. $\endgroup$ Commented Mar 21, 2022 at 18:51
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    $\begingroup$ @usεr11852 Does using transfer entropy for this lead one to conclude that chocolate rabbits and painted eggs cause Easter? $\endgroup$
    – dimitriy
    Commented Mar 21, 2022 at 18:53
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    $\begingroup$ The structural causal model for $x_{i}\to x_{j}$ is potentially different for every pair of variables (so the computational cost would likely be much higher than for a correlation matrix). If you are explicitly temporally ordering variable measures, you have the benefit of needing only the lower (or upper) triangle of the matrix; the matrix would otherwise be asymmetric since, unlike correlation, causation is directional. Finally, you need to think about what the diagonal means: some causal formalisms admit 'self-effects' (i.e. $x_i \to x_i$), and some do not. $\endgroup$
    – Alexis
    Commented Mar 21, 2022 at 18:54
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    $\begingroup$ @AdrianKeister: Apologies my do-Calculus understanding is rudimentary but as Richard pointed out, what you describe looks very much like a is a distribution-like object. We are looking for a scalar-like object; I guess taking a MAP could suffice? - but we would also need to define relevant priors. $\endgroup$
    – usεr11852
    Commented Mar 21, 2022 at 20:39

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