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This question already has an answer here:

Suppose you have some vector $[a_1, \ldots, a_n]$ where the terms are non-negative.

How do you call the normalization method where you divide each element by the sum (to get a probability vector)? i.e. $$a_i \mapsto \frac{a_i}{\sum a_j}$$

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marked as duplicate by whuber Mar 3 at 22:08

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  • $\begingroup$ This is not necessarily a probability vector -- are the $a_i$ constrained to be positive? $\endgroup$ – Sycorax Mar 3 at 18:18
  • $\begingroup$ @Sycorax, yes, sorry for not mentioning that $\endgroup$ – yaseco Mar 3 at 18:22
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You could call $a_i \mapsto \dfrac{a_i}{\sum\limits_j |a_j|}$ a Manhattan or $\ell_1$-normalisation, effectively projecting each point onto the boundary of a unit hyperoctahedron or cross-polytope, in rather the same way that $a_i \mapsto \dfrac{a_i}{\sqrt{\sum\limits_j a_j^2}}$ could be called a Euclidean or $\ell_2$-normalisation projecting each point onto the boundary of a unit hypersphere.

As you say, if the $a_i$ are all non-negative and at least one is positive, then this is equivalent to $a_i \mapsto \dfrac{a_i}{\sum\limits_j a_j}$ and once normalised they add up to $1$ (you are effectively restricting yourself to a single orthant simplex facet of the unit hyperoctahedron) so these could represent a probability distribution. In a sense you could be starting with values proportional to the probabilities, and the normalisation turns them into actual probabilities.

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