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The generalized Bernoulli distribution, aka the categorical distribution is a discrete probability distribution on $n$ exclusive events $(1, 2, \dots, n)$

A special case would be a distribution where the probability of one event would be 1 and all other events would have a probability of 0. I would like to know if there is some special name for such a probability distribution. I tried to look for singular or collapsed generalized Bernoulli distribution but I did not find anything relevant.

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    $\begingroup$ Do you have a reference for this being called Bernoulli? (As far as I know, Bernoulli distribution is the one where $X=1$ with probability $p$ and $X=0$ with probability $1-p$) $\endgroup$ – Juho Kokkala Oct 9 '17 at 17:34
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    $\begingroup$ Some common terms are "singular," "degenerate," "atomic," and "constant." Please note that the Bernoulli distribution is for a variable that takes on just $n=2$ values, $0$ and $1$. Perhaps you are thinking of a Binomial distribution? $\endgroup$ – whuber Oct 9 '17 at 17:34
  • $\begingroup$ @JuhoKokkala and whuber, yes, you're right, it is in fact generalized Bernoulli distribution (en.wikipedia.org/wiki/…). Thank you for the fix! $\endgroup$ – Martin Drozdik Oct 9 '17 at 17:36
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    $\begingroup$ one-hot? en.wikipedia.org/wiki/One-hot $\endgroup$ – Hugh Perkins Oct 9 '17 at 20:19
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You have a generalized Bernoulli distribution which can be described with a vector of probabilities $(p_1,p_2, \dotsc, p_k)$. You ask about the very special case $(0,0,\dotsc,1)$ which do not seem very interesting---with probability 1 the outcome will be $(0,0,\dotsc,1)$, so why do you even need a special name for this? What do you want to do with this very special and uninteresting distribution? Do not seem particularly useful. If you do have an interesting application, please do tell us!

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  • $\begingroup$ Imagine that you are trying to predict the outcome of a dice roll. Your prediction is summarized as a generalized Bernoulli distribution on the set of {1,2,3,4,5,6}. Then you roll the dice and see that the outcome is 4. How good was your prediction? One measure would be to compare your prediction distribution with this singular distribution that gives 4 with probability 1. $\endgroup$ – Martin Drozdik May 4 '19 at 13:48
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    $\begingroup$ Martin That application would not inspire anyone to give a special name to the degenerate distribution, though. Indeed, it strongly suggests giving it a special name would be unhelpful, because it views all these distributions--even the degenerate ones--as belonging to a common family. $\endgroup$ – whuber Oct 15 '19 at 13:36

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