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Both seem to result in one of k different separated outcomes, and Wikipedia says these are often conflated. Despite reading the explanation of the difference on the article about multinomial distribution, I still have trouble understanding what the difference really is.

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The multinomial distribution is when there are multiple identical independent trials where each trial has $k$ possible outcomes.

The categorical distribution is when there is only one such trial.

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    $\begingroup$ +1. It corresponds the distinction between the binomial and the Bernoulli distributions: a sample size of $n$ compared with a sample size of $1$ $\endgroup$ – Henry Oct 10 '14 at 10:41
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    $\begingroup$ Multinomial : Binomial :: Categorical : Bernoulli $\endgroup$ – Leopd Apr 12 '15 at 6:30
  • $\begingroup$ I thought the name multinomial was opposed to binomial. i.e. binomial distribution deals with two category case and multinomial distribution with several categories, in both cases having multiple trials. The answer somehow suggests to me that the multi part of multinomial refers to it dealing with >1 independent trials. $\endgroup$ – ddiez Jun 30 '15 at 4:51
  • $\begingroup$ Otherwise agree with the answer BTW. $\endgroup$ – ddiez Jun 30 '15 at 4:52
  • $\begingroup$ It is the multinational distribution with parameter n = 1. $\endgroup$ – Wei Zhong Aug 9 '18 at 23:42
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Think of it like this proportion:-

Bernoulli:Binomial::Categorical:Multinomial

So, just like Bernoulli distribution gives us the probability for a binary variable at each instance while Binomial returns it for N examples, Categorical distribution gives us the probability for a k-classifying variable at each instance while a Multinomial distribution returns it for N examples.

Thus: Categorical(Pk) = Multinomial(Pk, 1)

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