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According to Scikit's documentation, the Adjusted Rand Index (ARI) can be defined as:

$$\mathrm{ARI} = \frac{\mathrm{RI} - \mathbb{E}[\mathrm{RI}]}{\max(\mathrm{RI})-\mathbb{E}[\mathrm{RI}]}$$

I don't understand what the $\max(\mathrm{RI})$ refers to. The formula to calculate it can be found in Wikipedia for example, but I want to know intuitively why that value appears there instead of the total number of pairs as in the denominator of the $\mathrm{RI}$ (the not-adjusted version).

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The maximum possible RI, of a perfect result.

This is the usual adjustment for chance scaling, that maps the best possible result to 1, a random result to 0, and anything worse than random to less than 0.

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  • $\begingroup$ But isn't the maximum possible RI always $1$, by definition? In this case, why is the denominator $\max(\mathrm{RI}) - \mathbb{E}[\mathrm{RI}]$ instead of $1 - \mathbb{E}[\mathrm{RI}]$? $\endgroup$
    – Tendero
    Commented Nov 30, 2018 at 13:17
  • $\begingroup$ It should be 1 - until you cancel out the common terms. There is nothing wrong with writing max RI even if it is 1, if you want to emphasize that you need the maximum (but don't care if it is 1 for RI). The same equation can also be used, e.g., with MI then. $\endgroup$
    – Has QUIT--Anony-Mousse
    Commented Nov 30, 2018 at 17:59
  • $\begingroup$ I may sound stupid here, but what does a perfect result mean? If I'm comparing A and B, does the denominator denote the RI score if A looks exactly like B and vice versa? In that case, if I compare A and C, wouldn't the value be the same then (well this is not the case when i calculate it by hand). $\endgroup$
    – woof
    Commented Mar 18, 2020 at 0:12

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